[proofplan] The proof uses Oka's coherence theorem to reduce every assertion to finite local presentations by finite free $\mathcal{O}_X$-modules. First we prove a local quotient lemma: a quotient of a coherent module by a coherent submodule has a finite presentation. We then prove kernel stability by pulling a morphism back to a finite free presentation of its source and controlling the resulting relation sheaf with the coherence of the target. Image, cokernel, extension, restriction, and tensor product stability follow by explicitly writing the relevant finite presentations. [/proofplan]

[step:Use Oka's theorem to work with finite local presentations] Let $Y \subset X$ be an open complex submanifold, and let $\mathcal{O}_Y$ denote its sheaf of holomorphic functions. By Oka's Coherence Theorem, $\mathcal{O}_Y$ is coherent as an $\mathcal{O}_Y$-module. We use the finite-presentation criterion for coherent modules in the following form: an $\mathcal{O}_Y$-module $\mathcal{M}$ is coherent if and only if every point $y \in Y$ has an open neighborhood $V \subset Y$, nonnegative integers $a,b$, and an exact sequence \begin{align*} \mathcal{O}_V^a \xrightarrow{A} \mathcal{O}_V^b \xrightarrow{B} \mathcal{M}|_V \to 0 \end{align*} of $\mathcal{O}_V$-modules. We also use the defining consequence of coherence: if $\mathcal{N}$ is a coherent $\mathcal{O}_Y$-module and $C: \mathcal{O}_V^r \to \mathcal{N}|_V$ is an $\mathcal{O}_V$-module morphism from a finite free module, then $\ker C$ is locally finitely generated. Since coherence is local on the base, it is enough in each part to construct such a finite presentation after shrinking around an arbitrary point. [/step]

[step:Prove that quotients by coherent submodules are coherent]Let $\iota: \mathcal{A} \to \mathcal{B}$ be a monomorphism of $\mathcal{O}_X$-modules, and assume that $\mathcal{A}$ and $\mathcal{B}$ are coherent. We prove that $\mathcal{B}/\iota(\mathcal{A})$ is coherent. Fix $x \in X$. By coherence of $\mathcal{B}$, after replacing a neighborhood of $x$ by an open set $V \subset X$, there are nonnegative integers $n,m$ and an exact sequence \begin{align*} \mathcal{O}_V^n \xrightarrow{\sigma} \mathcal{O}_V^m \xrightarrow{\pi} \mathcal{B}|_V \to 0. \end{align*} By finite generation of $\mathcal{A}$, after shrinking $V$ again there is a nonnegative integer $q$ and an epimorphism \begin{align*} \eta: \mathcal{O}_V^q \to \mathcal{A}|_V. \end{align*} Since $\pi$ is an epimorphism of sheaves and $\mathcal{O}_V^q$ is finite free, we may shrink $V$ so that the composite $\iota \circ \eta: \mathcal{O}_V^q \to \mathcal{B}|_V$ lifts through $\pi$. Thus there is an $\mathcal{O}_V$-module morphism \begin{align*} \widetilde{\eta}: \mathcal{O}_V^q \to \mathcal{O}_V^m \end{align*} such that $\pi \circ \widetilde{\eta} = \iota \circ \eta$. Let $\overline{\pi}: \mathcal{O}_V^m \to \mathcal{B}|_V/\iota(\mathcal{A}|_V)$ be the composite of $\pi$ with the quotient morphism. The morphism $\overline{\pi}$ is an epimorphism. Its kernel is generated by $\operatorname{im}\sigma$ and $\operatorname{im}\widetilde{\eta}$, because a local section $s$ of $\mathcal{O}_V^m$ maps to zero in the quotient exactly when $\pi(s)$ locally lies in $\iota(\mathcal{A})$, and then a local lift through $\eta$ reduces $s$ modulo $\operatorname{im}\widetilde{\eta}$ to an element of $\ker \pi = \operatorname{im}\sigma$. Hence \begin{align*} \mathcal{O}_V^n \oplus \mathcal{O}_V^q \xrightarrow{\sigma \oplus \widetilde{\eta}} \mathcal{O}_V^m \xrightarrow{\overline{\pi}} \mathcal{B}|_V/\iota(\mathcal{A}|_V) \to 0 \end{align*} is exact, where $(\sigma \oplus \widetilde{\eta})(u,v)=\sigma(u)+\widetilde{\eta}(v)$. This is a finite presentation near $x$, so $\mathcal{B}/\iota(\mathcal{A})$ is coherent.[/step]

[guided]The point is to show that no infinite set of relations is introduced when we divide a coherent sheaf by a coherent subsheaf. Let $\iota: \mathcal{A} \to \mathcal{B}$ be an injective morphism of $\mathcal{O}_X$-modules, with both $\mathcal{A}$ and $\mathcal{B}$ coherent. We must produce a finite presentation for the quotient sheaf $\mathcal{B}/\iota(\mathcal{A})$ near an arbitrary point $x \in X$. Since $\mathcal{B}$ is coherent, the finite-presentation criterion gives an open neighborhood $V \subset X$ of $x$, nonnegative integers $n,m$, and an exact sequence \begin{align*} \mathcal{O}_V^n \xrightarrow{\sigma} \mathcal{O}_V^m \xrightarrow{\pi} \mathcal{B}|_V \to 0. \end{align*} Since $\mathcal{A}$ is coherent, it is locally finitely generated. After shrinking $V$, there is a nonnegative integer $q$ and an epimorphism \begin{align*} \eta: \mathcal{O}_V^q \to \mathcal{A}|_V. \end{align*} We now lift the chosen generators of $\mathcal{A}$ to the finite free module presenting $\mathcal{B}$. The composite \begin{align*} \iota \circ \eta: \mathcal{O}_V^q \to \mathcal{B}|_V \end{align*} is a morphism from a finite free module. Because $\pi$ is an epimorphism of sheaves, each of the finitely many standard basis sections of $\mathcal{O}_V^q$ lifts locally through $\pi$. Shrinking $V$ to the common domain of those finitely many lifts gives an $\mathcal{O}_V$-module morphism \begin{align*} \widetilde{\eta}: \mathcal{O}_V^q \to \mathcal{O}_V^m \end{align*} with $\pi \circ \widetilde{\eta} = \iota \circ \eta$. Let $\overline{\pi}: \mathcal{O}_V^m \to \mathcal{B}|_V/\iota(\mathcal{A}|_V)$ be the quotient of $\pi$. This map is surjective because $\pi$ is surjective. We identify its kernel. A local section $s$ of $\mathcal{O}_V^m$ lies in $\ker \overline{\pi}$ precisely when $\pi(s)$ lies locally in $\iota(\mathcal{A})$. Since $\eta$ is an epimorphism, this means that locally there is a section $t$ of $\mathcal{O}_V^q$ such that \begin{align*} \pi(s)=\iota(\eta(t))=\pi(\widetilde{\eta}(t)). \end{align*} Therefore $s-\widetilde{\eta}(t)$ lies in $\ker \pi$, and $\ker \pi=\operatorname{im}\sigma$ by exactness of the presentation of $\mathcal{B}$. Thus locally \begin{align*} s=\sigma(u)+\widetilde{\eta}(t) \end{align*} for some section $u$ of $\mathcal{O}_V^n$. This proves that \begin{align*} \ker \overline{\pi}=\operatorname{im}\sigma+\operatorname{im}\widetilde{\eta}. \end{align*} Hence the sequence \begin{align*} \mathcal{O}_V^n \oplus \mathcal{O}_V^q \xrightarrow{\sigma \oplus \widetilde{\eta}} \mathcal{O}_V^m \xrightarrow{\overline{\pi}} \mathcal{B}|_V/\iota(\mathcal{A}|_V) \to 0 \end{align*} is exact, where $(\sigma \oplus \widetilde{\eta})(u,t)=\sigma(u)+\widetilde{\eta}(t)$. This is a finite presentation near $x$. Since $x$ was arbitrary and coherence is local, $\mathcal{B}/\iota(\mathcal{A})$ is coherent.[/guided]

[step:Prove kernels of morphisms between coherent modules are coherent]Let $\phi: \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-module morphism. Fix $x \in X$. Since $\mathcal{F}$ is coherent, after shrinking to an open neighborhood $V \subset X$ of $x$ there are nonnegative integers $q,r$ and an exact sequence \begin{align*} \mathcal{O}_V^q \xrightarrow{\alpha} \mathcal{O}_V^r \xrightarrow{\beta} \mathcal{F}|_V \to 0. \end{align*} Define \begin{align*} \psi: \mathcal{O}_V^r \to \mathcal{G}|_V \end{align*} by $\psi=\phi|_V \circ \beta$, and set $\mathcal{H}:=\ker \psi$. Since $\mathcal{G}$ is coherent and $\mathcal{O}_V^r$ is finite free, $\mathcal{H}$ is locally finitely generated. After shrinking $V$, choose an epimorphism \begin{align*} \theta: \mathcal{O}_V^m \to \mathcal{H} \end{align*} for some nonnegative integer $m$. Let $j: \mathcal{H} \to \mathcal{O}_V^r$ be the inclusion. The kernel of $j \circ \theta: \mathcal{O}_V^m \to \mathcal{O}_V^r$ is locally finitely generated by Oka coherence, because this is a morphism between finite free modules. After shrinking $V$, choose an epimorphism \begin{align*} \lambda: \mathcal{O}_V^n \to \ker(j \circ \theta). \end{align*} Then \begin{align*} \mathcal{O}_V^n \xrightarrow{\lambda} \mathcal{O}_V^m \xrightarrow{\theta} \mathcal{H} \to 0 \end{align*} is a finite presentation, so $\mathcal{H}$ is coherent. Because $\beta \circ \alpha=0$, we have $\psi \circ \alpha=0$, so $\operatorname{im}\alpha \subset \mathcal{H}$. The morphism $\alpha: \mathcal{O}_V^q \to \mathcal{O}_V^r$ is a morphism between finite free modules, so Oka coherence gives a finite local presentation of $\operatorname{im}\alpha$; after shrinking, $\operatorname{im}\alpha$ is coherent. By the quotient result just proved, $\mathcal{H}/\operatorname{im}\alpha$ is coherent. The morphism $\beta$ induces an isomorphism \begin{align*} \mathcal{H}/\operatorname{im}\alpha \to (\ker \phi)|_V. \end{align*} Indeed, a local section $h$ of $\mathcal{H}$ satisfies $\phi(\beta(h))=\psi(h)=0$, so $\beta(h)$ lies in $\ker \phi$; local surjectivity follows from the epimorphism $\beta$; and the kernel is exactly $\ker\beta=\operatorname{im}\alpha$. Hence $(\ker \phi)|_V$ is coherent near $x$. Since $x$ was arbitrary, $\ker \phi$ is coherent.[/step]

[guided]We want to prove that the kernel of $\phi: \mathcal{F} \to \mathcal{G}$ is coherent. The strategy is to replace $\mathcal{F}$ by a finite free presentation and then describe $\ker \phi$ as a quotient of a coherent subsheaf of that finite free module. Fix a point $x \in X$. Since $\mathcal{F}$ is coherent, the finite-presentation criterion gives, after shrinking to an open neighborhood $V \subset X$ of $x$, nonnegative integers $q,r$ and an exact sequence \begin{align*} \mathcal{O}_V^q \xrightarrow{\alpha} \mathcal{O}_V^r \xrightarrow{\beta} \mathcal{F}|_V \to 0. \end{align*} Compose the presentation map $\beta$ with $\phi$. Define the $\mathcal{O}_V$-module morphism \begin{align*} \psi: \mathcal{O}_V^r \to \mathcal{G}|_V \end{align*} by $\psi=\phi|_V \circ \beta$, and define the subsheaf \begin{align*} \mathcal{H}:=\ker \psi \subset \mathcal{O}_V^r. \end{align*} The coherence of $\mathcal{G}$ is now used in its defining form: the kernel of a morphism from a finite free module into a coherent module is locally finitely generated. Since $\mathcal{O}_V^r$ is finite free and $\mathcal{G}|_V$ is coherent, $\mathcal{H}$ is locally finitely generated. After shrinking $V$, there is a nonnegative integer $m$ and an epimorphism \begin{align*} \theta: \mathcal{O}_V^m \to \mathcal{H}. \end{align*} Let $j: \mathcal{H} \to \mathcal{O}_V^r$ be the inclusion morphism. The composite \begin{align*} j \circ \theta: \mathcal{O}_V^m \to \mathcal{O}_V^r \end{align*} is a morphism between finite free $\mathcal{O}_V$-modules. By Oka coherence of $\mathcal{O}_V$, the kernel of this morphism is locally finitely generated. After shrinking $V$ again, choose an epimorphism \begin{align*} \lambda: \mathcal{O}_V^n \to \ker(j \circ \theta) \end{align*} for some nonnegative integer $n$. Since $\ker(j \circ \theta)=\ker \theta$, we have an exact sequence \begin{align*} \mathcal{O}_V^n \xrightarrow{\lambda} \mathcal{O}_V^m \xrightarrow{\theta} \mathcal{H} \to 0. \end{align*} This is a finite presentation of $\mathcal{H}$, so $\mathcal{H}$ is coherent. Now compare $\mathcal{H}$ with $\ker \phi$. The relation sheaf $\operatorname{im}\alpha$ lies inside $\mathcal{H}$ because \begin{align*} \psi \circ \alpha = \phi|_V \circ \beta \circ \alpha = 0. \end{align*} We also need $\operatorname{im}\alpha$ itself to be coherent before taking a quotient. The map $\alpha: \mathcal{O}_V^q \to \mathcal{O}_V^r$ is a morphism between finite free modules, so Oka coherence implies that $\ker \alpha$ is locally finitely generated. After shrinking, this gives a finite presentation of $\operatorname{im}\alpha$, hence $\operatorname{im}\alpha$ is coherent. By the quotient result, $\mathcal{H}/\operatorname{im}\alpha$ is coherent. Finally, the presentation of $\mathcal{F}$ identifies this quotient with $(\ker \phi)|_V$. The induced morphism \begin{align*} \mathcal{H}/\operatorname{im}\alpha \to (\ker \phi)|_V \end{align*} sends the class of a local section $h$ of $\mathcal{H}$ to $\beta(h)$. This is well-defined because $\beta(\operatorname{im}\alpha)=0$. It lands in $\ker \phi$ because \begin{align*} \phi(\beta(h))=\psi(h)=0. \end{align*} It is locally surjective because every local section of $\mathcal{F}|_V$ locally lifts through the epimorphism $\beta$, and a lift of a section killed by $\phi$ lies in $\ker \psi=\mathcal{H}$. Its kernel consists exactly of those $h \in \mathcal{H}$ with $\beta(h)=0$, namely $\ker\beta=\operatorname{im}\alpha$. Therefore \begin{align*} \mathcal{H}/\operatorname{im}\alpha \cong (\ker \phi)|_V. \end{align*} Thus $\ker \phi$ is coherent near $x$, and since $x$ was arbitrary, $\ker \phi$ is coherent on $X$.[/guided]

[step:Deduce coherence of the image and cokernel from kernels and quotients] For the image, the exact sequence \begin{align*} 0 \to \ker \phi \to \mathcal{F} \to \operatorname{im}\phi \to 0 \end{align*} identifies $\operatorname{im}\phi$ with $\mathcal{F}/\ker\phi$. The kernel is coherent by the previous step, and $\mathcal{F}$ is coherent by hypothesis. Therefore the quotient result gives that $\operatorname{im}\phi$ is coherent. For the cokernel, the exact sequence \begin{align*} 0 \to \operatorname{im}\phi \to \mathcal{G} \to \operatorname{coker}\phi \to 0 \end{align*} identifies $\operatorname{coker}\phi$ with $\mathcal{G}/\operatorname{im}\phi$. Since $\operatorname{im}\phi$ and $\mathcal{G}$ are coherent, the quotient result gives that $\operatorname{coker}\phi$ is coherent. [/step]

[step:Prove the extension statement by writing a presentation in the remaining case]Let \begin{align*} 0 \to \mathcal{F}' \xrightarrow{i} \mathcal{F} \xrightarrow{p} \mathcal{F}'' \to 0 \end{align*} be an exact sequence of $\mathcal{O}_X$-modules. If $\mathcal{F}'$ and $\mathcal{F}$ are coherent, then $\mathcal{F}'' \cong \mathcal{F}/i(\mathcal{F}')$ is coherent by the quotient result. If $\mathcal{F}$ and $\mathcal{F}''$ are coherent, then $\mathcal{F}' \cong \ker p$ is coherent by kernel stability. It remains to assume that $\mathcal{F}'$ and $\mathcal{F}''$ are coherent and prove that $\mathcal{F}$ is coherent. Fix $x \in X$. After shrinking to an open neighborhood $V \subset X$ of $x$, choose finite presentations \begin{align*} \mathcal{O}_V^d \xrightarrow{\nu} \mathcal{O}_V^c \xrightarrow{\eta} \mathcal{F}'|_V \to 0 \end{align*} and \begin{align*} \mathcal{O}_V^a \xrightarrow{\rho} \mathcal{O}_V^b \xrightarrow{\beta} \mathcal{F}''|_V \to 0. \end{align*} Since $p|_V$ is an epimorphism of sheaves and $\mathcal{O}_V^b$ is finite free, shrink $V$ so that there is an $\mathcal{O}_V$-module morphism \begin{align*} s: \mathcal{O}_V^b \to \mathcal{F}|_V \end{align*} with $p \circ s=\beta$. Define \begin{align*} H: \mathcal{O}_V^c \oplus \mathcal{O}_V^b &\to \mathcal{F}|_V \\ (u,v) &\mapsto i(\eta(u))+s(v). \end{align*} The morphism $H$ is an epimorphism: any local section of $\mathcal{F}|_V$ maps under $p$ to a local section of $\mathcal{F}''|_V$, which locally lifts through $\beta$, and after subtracting the corresponding value of $s$ the remaining section lies in $i(\mathcal{F}'|_V)$ and locally lifts through $\eta$. Since $\beta \circ \rho=0$, we have $p \circ s \circ \rho=0$. Exactness gives a unique morphism \begin{align*} t: \mathcal{O}_V^a \to \mathcal{F}'|_V \end{align*} such that $i \circ t=s \circ \rho$. Shrinking $V$ again, lift $t$ through $\eta$ to an $\mathcal{O}_V$-module morphism \begin{align*} \ell: \mathcal{O}_V^a \to \mathcal{O}_V^c \end{align*} with $\eta \circ \ell=t$. Define \begin{align*} Q: \mathcal{O}_V^d \oplus \mathcal{O}_V^a &\to \mathcal{O}_V^c \oplus \mathcal{O}_V^b \\ (w,z) &\mapsto (\nu(w)-\ell(z),\rho(z)). \end{align*} Then $\operatorname{im}Q=\ker H$. The inclusion $\operatorname{im}Q \subset \ker H$ follows from $\eta\circ\nu=0$ and $i\circ\eta\circ\ell=s\circ\rho$. Conversely, if $(u,v)$ is a local section of $\ker H$, then applying $p$ gives $\beta(v)=0$, so locally $v=\rho(z)$. Then \begin{align*} 0=H(u,\rho(z))=i(\eta(u)+t(z)). \end{align*} Since $i$ is a monomorphism, $\eta(u)+t(z)=0$. As $t=\eta\circ\ell$, this means $\eta(u+\ell(z))=0$, so locally $u+\ell(z)=\nu(w)$. Hence $(u,v)=Q(w,z)$. Therefore \begin{align*} \mathcal{O}_V^d \oplus \mathcal{O}_V^a \xrightarrow{Q} \mathcal{O}_V^c \oplus \mathcal{O}_V^b \xrightarrow{H} \mathcal{F}|_V \to 0 \end{align*} is a finite presentation. Thus $\mathcal{F}$ is coherent near $x$, and therefore coherent on $X$.[/step]

[guided]There are three possible pairs of coherent terms in the exact sequence \begin{align*} 0 \to \mathcal{F}' \xrightarrow{i} \mathcal{F} \xrightarrow{p} \mathcal{F}'' \to 0. \end{align*} Two are immediate from previous results. If $\mathcal{F}'$ and $\mathcal{F}$ are coherent, then $\mathcal{F}''$ is the quotient $\mathcal{F}/i(\mathcal{F}')$, so the quotient lemma gives coherence of $\mathcal{F}''$. If $\mathcal{F}$ and $\mathcal{F}''$ are coherent, then $\mathcal{F}'$ is the kernel of $p$, so kernel stability gives coherence of $\mathcal{F}'$. The only case requiring a new construction is when $\mathcal{F}'$ and $\mathcal{F}''$ are coherent and we must prove that the middle term $\mathcal{F}$ is coherent. Fix $x \in X$. Since $\mathcal{F}'$ and $\mathcal{F}''$ are coherent, after shrinking to an open neighborhood $V \subset X$ of $x$ we may choose finite presentations \begin{align*} \mathcal{O}_V^d \xrightarrow{\nu} \mathcal{O}_V^c \xrightarrow{\eta} \mathcal{F}'|_V \to 0 \end{align*} and \begin{align*} \mathcal{O}_V^a \xrightarrow{\rho} \mathcal{O}_V^b \xrightarrow{\beta} \mathcal{F}''|_V \to 0. \end{align*} We need generators for $\mathcal{F}$. The generators of $\mathcal{F}'$ already give elements of $\mathcal{F}$ through $i$. For the quotient $\mathcal{F}''$, we lift its chosen generators through $p$. Since $p|_V$ is an epimorphism of sheaves and $\mathcal{O}_V^b$ is finite free, after shrinking $V$ there is an $\mathcal{O}_V$-module morphism \begin{align*} s: \mathcal{O}_V^b \to \mathcal{F}|_V \end{align*} with $p\circ s=\beta$. Define \begin{align*} H: \mathcal{O}_V^c \oplus \mathcal{O}_V^b &\to \mathcal{F}|_V \\ (u,v) &\mapsto i(\eta(u))+s(v). \end{align*} This map is surjective. Indeed, let $f$ be a local section of $\mathcal{F}|_V$. Its image $p(f)$ is a local section of $\mathcal{F}''|_V$, so locally $p(f)=\beta(v)$ for some section $v$ of $\mathcal{O}_V^b$. Then \begin{align*} p(f-s(v))=p(f)-\beta(v)=0. \end{align*} Exactness gives $f-s(v)$ locally in $i(\mathcal{F}'|_V)$, and since $\eta$ is surjective, locally $f-s(v)=i(\eta(u))$ for some section $u$ of $\mathcal{O}_V^c$. Hence $f=H(u,v)$ locally. It remains to describe the relations among these generators. Relations in $\mathcal{F}'$ are generated by $\nu$. Relations in $\mathcal{F}''$ are generated by $\rho$, but each relation $\rho(z)$ among quotient generators lifts to an element $s(\rho(z))$ in $\mathcal{F}$ lying inside $i(\mathcal{F}')$. Because $p\circ s\circ\rho=\beta\circ\rho=0$, exactness gives a unique morphism \begin{align*} t: \mathcal{O}_V^a \to \mathcal{F}'|_V \end{align*} such that $i\circ t=s\circ\rho$. Since $\eta$ is an epimorphism and $\mathcal{O}_V^a$ is finite free, after shrinking $V$ we may lift $t$ to a morphism \begin{align*} \ell: \mathcal{O}_V^a \to \mathcal{O}_V^c \end{align*} with $\eta\circ\ell=t$. Define the relation map \begin{align*} Q: \mathcal{O}_V^d \oplus \mathcal{O}_V^a &\to \mathcal{O}_V^c \oplus \mathcal{O}_V^b \\ (w,z) &\mapsto (\nu(w)-\ell(z),\rho(z)). \end{align*} We check that $\operatorname{im}Q=\ker H$. First, \begin{align*} H(\nu(w)-\ell(z),\rho(z)) = i(\eta(\nu(w))-\eta(\ell(z)))+s(\rho(z)). \end{align*} Since $\eta\circ\nu=0$, $\eta\circ\ell=t$, and $i\circ t=s\circ\rho$, this becomes \begin{align*} 0-i(t(z))+i(t(z))=0. \end{align*} Thus $\operatorname{im}Q \subset \ker H$. Conversely, suppose $(u,v)$ is a local section of $\ker H$. Applying $p$ gives \begin{align*} 0=p(H(u,v))=\beta(v). \end{align*} Since $\ker\beta=\operatorname{im}\rho$, locally $v=\rho(z)$ for some section $z$ of $\mathcal{O}_V^a$. Then \begin{align*} 0=H(u,\rho(z))=i(\eta(u))+s(\rho(z))=i(\eta(u)+t(z)). \end{align*} Because $i$ is a monomorphism, $\eta(u)+t(z)=0$. Since $t=\eta\circ\ell$, we get \begin{align*} \eta(u+\ell(z))=0. \end{align*} As $\ker\eta=\operatorname{im}\nu$, locally $u+\ell(z)=\nu(w)$ for some section $w$ of $\mathcal{O}_V^d$. Therefore \begin{align*} (u,v)=(\nu(w)-\ell(z),\rho(z))=Q(w,z). \end{align*} So $\ker H=\operatorname{im}Q$. We have constructed an exact sequence \begin{align*} \mathcal{O}_V^d \oplus \mathcal{O}_V^a \xrightarrow{Q} \mathcal{O}_V^c \oplus \mathcal{O}_V^b \xrightarrow{H} \mathcal{F}|_V \to 0. \end{align*} This is a finite presentation of $\mathcal{F}$ near $x$. Since $x$ was arbitrary, $\mathcal{F}$ is coherent on $X$.[/guided]

[step:Restrict finite presentations to open subsets] Let $U \subset X$ be open, and assume $\mathcal{F}$ is coherent on $X$. Fix $x \in U$. Choose an open neighborhood $V \subset X$ of $x$, nonnegative integers $a,b$, and an exact sequence \begin{align*} \mathcal{O}_V^a \xrightarrow{A} \mathcal{O}_V^b \xrightarrow{B} \mathcal{F}|_V \to 0. \end{align*} Set $W:=U\cap V$. Restricting the exact sequence to $W$ gives \begin{align*} \mathcal{O}_W^a \xrightarrow{A|_W} \mathcal{O}_W^b \xrightarrow{B|_W} \mathcal{F}|_W \to 0, \end{align*} because restriction of sheaves preserves exactness and $\mathcal{O}_V|_W=\mathcal{O}_W$. Thus $\mathcal{F}|_U$ is locally finitely presented over $\mathcal{O}_U$, so $\mathcal{F}|_U$ is coherent on $U$. [/step]

[step:Present the tensor product from presentations of the two factors]Fix $x \in X$. Since $\mathcal{F}$ and $\mathcal{G}$ are coherent, after shrinking to an open neighborhood $V \subset X$ of $x$ choose finite presentations \begin{align*} \mathcal{O}_V^a \xrightarrow{\alpha} \mathcal{O}_V^b \xrightarrow{\beta} \mathcal{F}|_V \to 0 \end{align*} and \begin{align*} \mathcal{O}_V^c \xrightarrow{\gamma} \mathcal{O}_V^d \xrightarrow{\delta} \mathcal{G}|_V \to 0. \end{align*} By the presentation formula for tensor products and the right exactness of tensor product, the morphism \begin{align*} P: (\mathcal{O}_V^a \otimes_{\mathcal{O}_V} \mathcal{O}_V^d) \oplus (\mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^c) \to \mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^d \end{align*} defined on local pure tensors by \begin{align*} P(u\otimes v,w\otimes z)=\alpha(u)\otimes v+w\otimes\gamma(z) \end{align*} fits into an exact sequence \begin{align*} (\mathcal{O}_V^a \otimes_{\mathcal{O}_V} \mathcal{O}_V^d) \oplus (\mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^c) \xrightarrow{P} \mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^d \xrightarrow{\beta\otimes\delta} (\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G})|_V \to 0. \end{align*} The tensor products of finite free $\mathcal{O}_V$-modules are finite free $\mathcal{O}_V$-modules, with ranks $ad$, $bc$, and $bd$ after choosing bases. Hence this is a finite presentation of $(\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G})|_V$. Therefore $\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G}$ is coherent.[/step]

[guided]We prove tensor stability by writing down the presentation rather than appealing only to an abstract closure property. Fix $x \in X$. Since $\mathcal{F}$ and $\mathcal{G}$ are coherent, after shrinking to an open neighborhood $V \subset X$ of $x$ we have finite presentations \begin{align*} \mathcal{O}_V^a \xrightarrow{\alpha} \mathcal{O}_V^b \xrightarrow{\beta} \mathcal{F}|_V \to 0 \end{align*} and \begin{align*} \mathcal{O}_V^c \xrightarrow{\gamma} \mathcal{O}_V^d \xrightarrow{\delta} \mathcal{G}|_V \to 0. \end{align*} The tensor product $\mathcal{F}|_V\otimes_{\mathcal{O}_V}\mathcal{G}|_V$ is generated by tensors of the chosen generators of $\mathcal{F}|_V$ and $\mathcal{G}|_V$. Thus the natural epimorphism is \begin{align*} \beta\otimes\delta: \mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^d \to (\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G})|_V. \end{align*} The relations come from the relations in either factor. A relation $\alpha(u)$ in the first factor gives $\alpha(u)\otimes v$, and a relation $\gamma(z)$ in the second factor gives $w\otimes\gamma(z)$. Therefore define \begin{align*} P: (\mathcal{O}_V^a \otimes_{\mathcal{O}_V} \mathcal{O}_V^d) \oplus (\mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^c) \to \mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^d \end{align*} by \begin{align*} P(u\otimes v,w\otimes z)=\alpha(u)\otimes v+w\otimes\gamma(z) \end{align*} on local pure tensors, extended $\mathcal{O}_V$-linearly. The presentation formula for tensor products is exactly the statement that these are all the relations; it follows from the right exactness of tensor product applied to the two finite presentations. Hence \begin{align*} (\mathcal{O}_V^a \otimes_{\mathcal{O}_V} \mathcal{O}_V^d) \oplus (\mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^c) \xrightarrow{P} \mathcal{O}_V^b \otimes_{\mathcal{O}_V} \mathcal{O}_V^d \xrightarrow{\beta\otimes\delta} (\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G})|_V \to 0 \end{align*} is exact. Since tensor products of finite free modules are finite free after choosing bases, this is a finite presentation. Therefore the tensor product is coherent near $x$, and since $x$ was arbitrary, $\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G}$ is coherent on $X$.[/guided]

[step:Combine the local conclusions to obtain all six assertions] Kernel stability was proved for every morphism $\phi:\mathcal{F}\to\mathcal{G}$. Image and cokernel coherence follow from kernel stability and the quotient lemma. The extension statement follows from the quotient case, the kernel case, and the explicit middle-term presentation. Restriction follows by restricting finite presentations, and tensor product stability follows from the tensor-product presentation. These are exactly the six assertions of the theorem. [/step]