In FOAG by Vakil, exercise 13.7.A part (b) I am supposed to show the following:
If $\mathcal{F,G}$ and $\mathcal{O}_X$ are coherent sheaves on a scheme $X$, then $\mathcal{Hom(F,G)}$ is also coherent.
I proceeded as following:
In part (a) I have shown that this sheaf Hom is quasicoherent. It therefore suffices to show, on an affine open $Spec(A)$, that if $A,M,N$ are coherent $A$ modules, then so is $Hom_A(M,N)$. Let there be exact sequence $A^{\oplus m} \to M\to 0$ since $M$ is finitely generated; using the left exactness of $Hom_A(\cdot, N)$, I get the exact sequence $0 \to Hom_A(M,N)\to Hom_A(A^{\oplus m},N) \cong N^{\oplus m}$. Now for any $A^{\oplus p} \to Hom_A(M,N) \subset N^{\oplus m}$, the kernel is finitely generated since $N^m$ is coherent. Therefore $Hom_A(M,N)$ is coherent.
I think this argument make sense, but I did not use the assumption that $\mathcal{O}_X$ is coherent. Am I missing something? Thanks!