[proofplan] We construct $X_0(N;n)$ as the modular curve representing elliptic curves equipped with two cyclic subgroup schemes of coprime orders $N$ and $n$ with trivial intersection. The condition $\gcd(n,N)=1$ ensures that the quotient by the $n$-subgroup carries a well-defined cyclic subgroup of order $N$, namely the image of $C$. The two forgetful and quotient constructions define morphisms to $X_0(N)$, and standard finite-level modular-curve representability and compactification theorems make these morphisms finite, hence a finite correspondence. [/proofplan]

[step:Construct the compact coarse modular curve from the Deligne-Rapoport moduli problem]Let $B:=\operatorname{Spec}\mathbb{Z}[1/Nn]$. A cyclic finite locally free subgroup scheme of rank $m$ means a finite locally free commutative subgroup scheme $G \subset E$ of rank $m$ which is fppf-locally isomorphic to the constant group scheme $\mathbb{Z}/m\mathbb{Z}$. Let $\mathscr{M}_0(N;n)$ denote the Deligne-Mumford stack over $B$ whose $S$-points are triples $(E,C,D)$, where $S$ is a $B$-scheme, $E \to S$ is an elliptic curve, $C \subset E$ is cyclic finite locally free of rank $N$, $D \subset E$ is cyclic finite locally free of rank $n$, and $C \cap D$ is the zero subgroup scheme of $E$. Since $Nn$ is invertible on $S$, both subgroup schemes are finite etale. On each geometric fibre, $C \cap D$ has order dividing both $N$ and $n$; since $\gcd(n,N)=1$, the fibre has order $1$. Thus the fibrewise condition is exactly the condition that $C \cap D$ is the zero subgroup scheme. We use the standard Deligne-Rapoport representability theorem in its coarse-moduli form: the stack of generalized elliptic curves equipped with ample cyclic subgroup schemes of the prescribed orders has a proper coarse modular curve over $B$, and its elliptic-curve locus is the coarse space of $\mathscr{M}_0(N;n)$. Here a Deligne-Rapoport compactification means this proper coarse curve obtained by allowing generalized elliptic curves at the boundary; compact modular curve means this proper coarse curve over $B$. We denote it by $X_0(N;n)$.[/step]

[guided]We first fix the base. Put $B:=\operatorname{Spec}\mathbb{Z}[1/Nn]$. The theorem is being proved over this base, so both $N$ and $n$ are invertible on every scheme under discussion. A cyclic finite locally free subgroup scheme of rank $m$ means a finite locally free commutative subgroup scheme $G \subset E$ of rank $m$ which is fppf-locally isomorphic to the constant group scheme $\mathbb{Z}/m\mathbb{Z}$. Define $\mathscr{M}_0(N;n)$ to be the Deligne-Mumford stack over $B$ whose $S$-points, for a $B$-scheme $S$, are triples $(E,C,D)$ with $E \to S$ an elliptic curve, $C \subset E$ cyclic finite locally free of rank $N$, $D \subset E$ cyclic finite locally free of rank $n$, and $C \cap D$ equal to the zero subgroup scheme of $E$. Why do we use a stack rather than the set-valued functor of isomorphism classes? Elliptic curves can have automorphisms preserving the level data, so the naive functor of isomorphism classes is not generally represented by a fine scheme. The correct representability statement is that the moduli problem is represented by a Deligne-Mumford stack and admits a coarse modular curve. This is the object meant by the modular curve in the theorem. The coprimality hypothesis verifies the intersection condition fibrewise. Since $Nn$ is invertible on $S$, the subgroup schemes $C$ and $D$ are finite etale. For a geometric point $s \to S$, the fibre $(C \cap D)_s$ is a subgroup of both $C_s$ and $D_s$, so its order divides both $N$ and $n$. Because $\gcd(n,N)=1$, this order is $1$. Thus every geometric fibre of $C \cap D$ is the identity subgroup, and the finite locally free intersection is the zero subgroup scheme. We now invoke the Deligne-Rapoport coarse-moduli theorem for modular curves with cyclic level structure. Its hypotheses are satisfied because the level structures are finite locally free cyclic subgroup schemes of fixed ranks over $B$, and at the compactification boundary the corresponding cyclic subgroups are required to be ample on generalized elliptic curves. The theorem supplies a proper coarse modular curve over $B$ whose open elliptic-curve locus is the coarse space of $\mathscr{M}_0(N;n)$. We call this proper coarse curve the Deligne-Rapoport compactification and denote it by $X_0(N;n)$.[/guided]

[step:Show the quotient construction preserves the $\Gamma_0(N)$ level structure]Let $(E,C,D)$ be an $S$-point of $X_0(N;n)$ over a scheme $S$, and let \begin{align*} q_D: E &\to E/D \end{align*} be the quotient isogeny with kernel $D$. Define \begin{align*} q_D(C) := (C + D)/D \subset E/D. \end{align*} The restriction $q_D|_C: C \to q_D(C)$ has kernel $C \cap D=0$, hence is an isomorphism of finite locally free group schemes. Therefore $q_D(C)$ is cyclic finite locally free of rank $N$. Thus $(E/D,q_D(C))$ is a point of $X_0(N)$.[/step]

[guided]Take an $S$-point $(E,C,D)$ of the moduli problem. The subgroup $D \subset E$ is finite locally free and cyclic of rank $n$, so the quotient elliptic curve $E/D$ exists, and we write the quotient isogeny as \begin{align*} q_D: E &\to E/D. \end{align*} The proposed second projection sends $(E,C,D)$ to the pair $(E/D,(C+D)/D)$. To justify this, we must prove that $(C+D)/D$ is a cyclic subgroup of $E/D$ of order $N$. Define \begin{align*} q_D(C) := (C+D)/D \subset E/D. \end{align*} The map $q_D|_C: C \to q_D(C)$ is a homomorphism of finite locally free group schemes. Its kernel is exactly $C \cap D$, because the elements of $C$ killed by the quotient map are precisely the elements of $C$ lying in $D$. By the defining condition of $X_0(N;n)$, this kernel is zero. Hence $q_D|_C$ is an isomorphism onto its image. Since $C$ is cyclic finite locally free of rank $N$, the image $q_D(C)$ is also cyclic finite locally free of rank $N$. Therefore $(E/D,q_D(C))$ has the required $\Gamma_0(N)$ level structure.[/guided]

[step:Define the two morphisms to $X_0(N)$] Define maps on $S$-points by \begin{align*} \pi_1(E,C,D) &= (E,C), \\ \pi_2(E,C,D) &= (E/D,(C+D)/D). \end{align*} The first assignment forgets $D$ and is compatible with base change. The second assignment is compatible with base change because quotients by finite locally free subgroup schemes and images of finite locally free subgroup schemes commute with base change in this setting. Hence both assignments are natural transformations from the functor represented by $X_0(N;n)$ to the functor represented by $X_0(N)$, and therefore induce morphisms of compact modular curves \begin{align*} \pi_1,\pi_2: X_0(N;n) &\to X_0(N). \end{align*} [/step]

[step:Extend the morphisms over the cusps]Let $Y_0(N)$ denote the open elliptic-curve locus of the compact coarse modular curve $X_0(N)$. On the open elliptic-curve locus, the preceding construction gives morphisms to $Y_0(N)$. For the boundary, use the Deligne-Rapoport compactification theorem in the following form: if a generalized elliptic curve is equipped with ample cyclic subgroup schemes of orders prime to the base, then forgetting one cyclic subgroup and quotienting by one cyclic subgroup define morphisms of the corresponding compact coarse modular curves. The hypotheses hold here because the compactification $X_0(N;n)$ is built from generalized elliptic curves with ample cyclic subgroup data of ranks $N$ and $n$, and $Nn$ is invertible on $B$. Moreover, since $C \cap D=0$, the restriction $q_D|_C$ remains a monomorphism on every geometric fibre, so the image of $C$ in the quotient is again an ample cyclic subgroup of rank $N$. Thus both $\pi_1$ and $\pi_2$ extend uniquely to morphisms $X_0(N;n) \to X_0(N)$.[/step]

[guided]Let $Y_0(N)$ be the open subcurve of $X_0(N)$ parameterising smooth elliptic curves with a cyclic subgroup of order $N$. Over the corresponding open locus in $X_0(N;n)$, the formulas for $\pi_1$ and $\pi_2$ have already been checked. The boundary consists of generalized elliptic curves, so we must use the compactified moduli problem, not only the smooth elliptic-curve argument. The Deligne-Rapoport compactification theorem says, in the form needed here, that compact modular curves with cyclic level structure are coarse moduli spaces for generalized elliptic curves equipped with ample cyclic subgroup schemes, and that the natural operations of forgetting a cyclic subgroup and quotienting by a cyclic subgroup define morphisms between these compact coarse curves. We verify the hypotheses. The base is $B=\operatorname{Spec}\mathbb{Z}[1/Nn]$, so the orders $N$ and $n$ are invertible on the base. By construction of $X_0(N;n)$, the boundary objects carry ample cyclic subgroup schemes $C$ and $D$ of ranks $N$ and $n$. The quotient operation is applied to the cyclic subgroup $D$, so the theorem gives a generalized elliptic curve quotient. The image of $C$ in that quotient is controlled by the same kernel computation as on the smooth locus: the kernel of $q_D|_C$ is $C \cap D$, which is the zero subgroup scheme. Hence $q_D(C)$ is isomorphic to $C$ and is cyclic of rank $N$. The ampleness condition is preserved under this isomorphism and quotient construction in the Deligne-Rapoport moduli problem. Therefore $\pi_1$ extends by forgetting $D$, while $\pi_2$ extends by quotienting by $D$ and taking the image of $C$. These extensions are morphisms of compact coarse modular curves $X_0(N;n) \to X_0(N)$.[/guided]

[step:Use properness and quasi-finiteness to obtain a finite correspondence]Both $X_0(N;n)$ and $X_0(N)$ are proper coarse modular curves over $B$, so the morphisms $\pi_1$ and $\pi_2$ are proper. For $\pi_1$, the geometric fibre over $(E,C)$ is the set of cyclic subgroup schemes $D \subset E[n]$ of rank $n$ satisfying $D \cap C=0$, which is finite because $E[n]$ is a finite group scheme. For $\pi_2$, a geometric point in the fibre determines a cyclic degree-$n$ isogeny onto the target elliptic curve or generalized elliptic curve; equivalently, it is controlled by a cyclic subgroup of the finite $n$-torsion of the source or dual target, so the fibre is finite. Thus both morphisms are quasi-finite. A proper quasi-finite morphism of schemes is finite, so $\pi_1$ and $\pi_2$ are finite. Therefore \begin{align*} X_0(N) \xleftarrow{\,\pi_1\,} X_0(N;n) \xrightarrow{\,\pi_2\,} X_0(N) \end{align*} is a finite correspondence on $X_0(N)$, and by definition this correspondence is the prime-to-level Hecke operator $T_n$.[/step]

[guided]To finish, we must prove finiteness of the two arrows. We use the standard criterion: a proper quasi-finite morphism of schemes is finite. First, the morphisms are proper. This follows because $X_0(N;n)$ and $X_0(N)$ are Deligne-Rapoport compact coarse modular curves over $B$, hence are proper over $B$, and the maps $\pi_1$ and $\pi_2$ are morphisms between these compactified moduli spaces. Now verify quasi-finiteness. For $\pi_1$, fix a geometric point $(E,C)$ of $X_0(N)$. A point in the fibre is exactly a cyclic subgroup scheme $D \subset E[n]$ of rank $n$ satisfying $D \cap C=0$. The group scheme $E[n]$ is finite because $n$ is invertible on the base, and a finite group scheme has only finitely many subgroup schemes on a geometric fibre. Hence the geometric fibres of $\pi_1$ are finite, so $\pi_1$ is quasi-finite. For $\pi_2$, fix a geometric target point $(E',C')$ of $X_0(N)$. A preimage consists of a cyclic degree-$n$ isogeny $E \to E'$ together with the compatible pullback of the $\Gamma_0(N)$ structure. Such isogenies are controlled by their cyclic kernels, or equivalently by cyclic subgroups of the finite $n$-torsion of the dual target. Since this finite group scheme has only finitely many cyclic subgroup schemes of rank $n$ on a geometric fibre, the fibre of $\pi_2$ is finite. Thus $\pi_2$ is quasi-finite. Properness plus quasi-finiteness gives that both $\pi_1$ and $\pi_2$ are finite. Therefore the diagram \begin{align*} X_0(N) \xleftarrow{\,\pi_1\,} X_0(N;n) \xrightarrow{\,\pi_2\,} X_0(N) \end{align*} defines a finite correspondence on $X_0(N)$, and this correspondence is denoted $T_n$.[/guided]