[proofplan] Work in abelian categories $\mathcal A$ and $\mathcal B$, so that short exact sequences, chain homology, and exactness of homology sequences are available. Choose projective resolutions of $A'$, $A$, and $A''$ that fit into a short exact sequence of chain complexes; this is supplied by the Horseshoe Lemma under the hypothesis that $\mathcal A$ has enough projectives. Because the short exact sequence of resolutions is termwise split, applying the additive functor $F$ preserves exactness degree by degree. The [long exact homology sequence](/theorems/4210) of the resulting short exact sequence of chain complexes gives the desired connecting morphisms and exactness. Finally, right exactness identifies $L_0F$ with $F$, and the dual statement follows by replacing projective resolutions and homology with injective resolutions and cohomology. [/proofplan] [step:Choose compatible projective resolutions] Let \begin{align*} 0 \to A' \xrightarrow{u} A \xrightarrow{v} A'' \to 0 \end{align*} be a short exact sequence in $\mathcal A$. Since $\mathcal A$ has enough projectives, choose projective resolutions \begin{align*} P'_\bullet &\to A',\\ P_\bullet &\to A,\\ P''_\bullet &\to A'', \end{align*} where each $P'_n$, $P_n$, and $P''_n$ is projective and the complexes are indexed in non-negative degrees. The ambient category $\mathcal A$ is abelian by the theorem statement and has enough projectives by hypothesis, so the hypotheses of the Horseshoe Lemma are satisfied for the displayed short exact sequence and the chosen projective resolutions of $A'$ and $A''$. By the Horseshoe Lemma, the resolution of $A$ may be chosen so that there is a short exact sequence of chain complexes \begin{align*} 0 \to P'_\bullet \xrightarrow{\alpha_\bullet} P_\bullet \xrightarrow{\beta_\bullet} P''_\bullet \to 0. \end{align*} Moreover, this sequence is termwise split: for every $n \geq 0$, \begin{align*} 0 \to P'_n \xrightarrow{\alpha_n} P_n \xrightarrow{\beta_n} P''_n \to 0 \end{align*} is a split short exact sequence in $\mathcal A$. [/step] [step:Apply the additive functor to obtain a short exact sequence of complexes] Apply the additive functor \begin{align*} F: \mathcal A \to \mathcal B \end{align*} degree by degree to the short exact sequence of projective resolutions. For each $n \geq 0$, the sequence \begin{align*} 0 \to P'_n \xrightarrow{\alpha_n} P_n \xrightarrow{\beta_n} P''_n \to 0 \end{align*} is split exact, so there is an isomorphism $P_n \cong P'_n \oplus P''_n$ compatible with $\alpha_n$ and $\beta_n$. Since $F$ is additive, it preserves finite biproduct decompositions, hence \begin{align*} F(P_n) \cong F(P'_n) \oplus F(P''_n). \end{align*} Therefore \begin{align*} 0 \to F(P'_n) \xrightarrow{F(\alpha_n)} F(P_n) \xrightarrow{F(\beta_n)} F(P''_n) \to 0 \end{align*} is split exact in $\mathcal B$ for every $n \geq 0$. Because $\alpha_\bullet$ and $\beta_\bullet$ are chain maps, applying $F$ gives chain maps \begin{align*} F(\alpha_\bullet): F(P'_\bullet) \to F(P_\bullet), \qquad F(\beta_\bullet): F(P_\bullet) \to F(P''_\bullet). \end{align*} Thus we obtain a short exact sequence of chain complexes in $\mathcal B$: \begin{align*} 0 \to F(P'_\bullet) \to F(P_\bullet) \to F(P''_\bullet) \to 0. \end{align*} [/step] [step:Take homology to get the connecting morphisms and exactness] Because $\mathcal B$ is abelian, the homology objects of chain complexes in $\mathcal B$ are defined, and the long exact homology sequence applies to every short exact sequence of chain complexes in $\mathcal B$. Since \begin{align*} 0 \to F(P'_\bullet) \to F(P_\bullet) \to F(P''_\bullet) \to 0 \end{align*} is short exact in the abelian category $\mathcal B$, there are connecting morphisms \begin{align*} \partial_i: H_i(F(P''_\bullet)) \to H_{i-1}(F(P'_\bullet)) \end{align*} for every $i \geq 1$, and the sequence \begin{align*} \cdots \to H_i(F(P'_\bullet)) \to H_i(F(P_\bullet)) \to H_i(F(P''_\bullet)) \xrightarrow{\partial_i} H_{i-1}(F(P'_\bullet)) \to \cdots \end{align*} is exact in $\mathcal B$. We use homological indexing for projective resolutions: the differential on $P_\bullet$ lowers degree, and the $i$-th left derived functor is defined by applying $F$ to a projective resolution and taking homology in degree $i$. Thus \begin{align*} L_iF(A') &= H_i(F(P'_\bullet)),\\ L_iF(A) &= H_i(F(P_\bullet)),\\ L_iF(A'') &= H_i(F(P''_\bullet)). \end{align*} Substituting these identifications gives \begin{align*} \cdots \to L_iF(A') \to L_iF(A) \to L_iF(A'') \xrightarrow{\partial_i} L_{i-1}F(A') \to \cdots \to L_0F(A') \to L_0F(A) \to L_0F(A'') \to 0. \end{align*} [/step] [step:Identify the degree zero term using right exactness] It remains to justify the terminal right exact part. For an object $X \in \mathcal A$, choose a projective resolution \begin{align*} \cdots \to P_1 \xrightarrow{d_1} P_0 \xrightarrow{\varepsilon} X \to 0. \end{align*} Since $F$ is right exact, the sequence \begin{align*} F(P_1) \xrightarrow{F(d_1)} F(P_0) \xrightarrow{F(\varepsilon)} F(X) \to 0 \end{align*} is exact. Hence \begin{align*} L_0F(X) = H_0(F(P_\bullet)) = \operatorname{coker}\bigl(F(d_1):F(P_1)\to F(P_0)\bigr) \cong F(X). \end{align*} Applying this natural identification to $X=A'$, $A$, and $A''$ identifies the final segment with \begin{align*} L_0F(A') \to L_0F(A) \to L_0F(A'') \to 0, \end{align*} and this is exact by the long exact homology sequence above. Thus the stated long exact sequence of left derived functors is obtained. [/step] [step:Lift morphisms of short exact sequences to compatible chain maps] Let \begin{align*} 0 \to A'_1 \to A_1 \to A''_1 \to 0 \end{align*} and \begin{align*} 0 \to A'_2 \to A_2 \to A''_2 \to 0 \end{align*} be short exact sequences in $\mathcal A$, and let a morphism from the first sequence to the second be given by morphisms \begin{align*} f':A'_1 \to A'_2,\qquad f:A_1 \to A_2,\qquad f'':A''_1 \to A''_2 \end{align*} commuting with the short exact sequence maps. Choose compatible projective resolutions for both short exact sequences as above. We use the comparison theorem in its form for short exact sequences of projective resolutions: given a morphism of short exact sequences in $\mathcal A$ and two Horseshoe constructions over those sequences, there exist chain maps \begin{align*} \phi'_\bullet:P'_{1,\bullet} \to P'_{2,\bullet},\qquad \phi_\bullet:P_{1,\bullet} \to P_{2,\bullet},\qquad \phi''_\bullet:P''_{1,\bullet} \to P''_{2,\bullet} \end{align*} lifting $f'$, $f$, and $f''$, respectively, such that the diagram of short exact sequences of chain complexes commutes. This compatible comparison statement follows from the ordinary comparison theorem for projective resolutions, using the lifting property of projective objects at each degree and correcting inductively so that the maps commute with the inclusions and projections of the Horseshoe sequences. [/step] [step:Use functoriality of the homology sequence to prove naturality] Applying $F$ degree by degree to the compatible chain maps gives a morphism of short exact sequences of chain complexes in $\mathcal B$. The long exact homology sequence in an abelian category is functorial with respect to morphisms of short exact sequences of chain complexes. Therefore the induced maps on homology commute with all connecting morphisms $\partial_i$. After identifying homology with the derived functors $L_iF$, this says precisely that the long exact sequence constructed above is natural in the original short exact sequence. If a different compatible system of comparison maps is chosen, the comparison theorem gives chain homotopic maps on each resolution, so the induced maps on homology are the same; hence the natural long exact sequence is independent of the auxiliary comparison choices. [/step] [step:Dualize the construction for right derived functors] Assume now that $\mathcal A$ has enough injectives and that \begin{align*} G:\mathcal A \to \mathcal B \end{align*} is additive and left exact. For the short exact sequence \begin{align*} 0 \to A' \to A \to A'' \to 0, \end{align*} choose compatible injective resolutions \begin{align*} A' &\to I'^\bullet,\\ A &\to I^\bullet,\\ A'' &\to I''^\bullet. \end{align*} The category $\mathcal A$ is abelian and has enough injectives by the dual hypothesis, so the hypotheses of the Dual Horseshoe Lemma for injective resolutions are satisfied. By that lemma, these may be chosen so that \begin{align*} 0 \to I'^\bullet \to I^\bullet \to I''^\bullet \to 0 \end{align*} is a short exact sequence of cochain complexes whose degree-$n$ component \begin{align*} 0 \to I'^n \to I^n \to I''^n \to 0 \end{align*} is split exact in $\mathcal A$ for every $n \geq 0$. Since $G$ is additive, applying $G$ degree by degree preserves the termwise split exactness, giving a short exact sequence of cochain complexes \begin{align*} 0 \to G(I'^\bullet) \to G(I^\bullet) \to G(I''^\bullet) \to 0. \end{align*} Because $\mathcal B$ is abelian, the [long exact cohomology sequence](/theorems/3471) applies to this short exact sequence of cochain complexes and gives connecting morphisms \begin{align*} \delta^i: H^i(G(I''^\bullet)) \to H^{i+1}(G(I'^\bullet)). \end{align*} We use cohomological indexing for injective resolutions: the differential on $I^\bullet$ raises degree, and the $i$-th right derived functor is defined by applying $G$ to an injective resolution and taking cohomology in degree $i$. Thus \begin{align*} R^iG(A') &= H^i(G(I'^\bullet)),\\ R^iG(A) &= H^i(G(I^\bullet)),\\ R^iG(A'') &= H^i(G(I''^\bullet)). \end{align*} Thus we obtain the exact sequence \begin{align*} 0 \to R^0G(A') \to R^0G(A) \to R^0G(A'') \xrightarrow{\delta^0} R^1G(A') \to R^1G(A) \to R^1G(A'') \xrightarrow{\delta^1} R^2G(A') \to \cdots. \end{align*} Left exactness gives the natural identification $R^0G \cong G$. For functoriality, use the compatible comparison theorem for short exact sequences of injective resolutions: a morphism of short exact sequences in $\mathcal A$ lifts to a morphism of the corresponding short exact sequences of cochain complexes, uniquely up to cochain homotopy on cohomology. Applying $G$ gives a morphism of short exact sequences of cochain complexes in $\mathcal B$, and the long exact cohomology sequence is functorial with respect to such morphisms. Therefore the connecting morphisms $\delta^i$ commute with the maps induced by morphisms of short exact sequences, and the construction is natural. This proves the dual statement. [/step]