Why is sheaf Hom left-exact?

Question

Fix a ringed space $(X,\mathcal{O}_X)$ and two sheaves of $\mathcal{O}_X$-modules $\mathscr{F}$ and $\mathscr{G}$, then we can define the sheaf $\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F},\mathscr{G})$ by setting $$\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F},\mathscr{G})(U)=\mathrm{Hom}_{\mathcal{O}_X\vert_U}(\mathscr{F}\vert_U,\mathscr{G}\vert_U)$$ with the obvious restriction maps, so that using maps of $\mathcal{O}_X$-modules $\varphi:\mathscr{G}\to\mathscr{G}'$ and $\psi:\mathscr{F}'\to\mathscr{F}$ we can induce maps $$\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F},\mathscr{G})\to \mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F},\mathscr{G}')$$ and $$\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F},\mathscr{G})\to\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F'},\mathscr{G})$$ given on the open set $U$ by $$f\mapsto \varphi\vert_U\circ f$$ and $$f\mapsto f\circ\psi\vert_U$$ respectively. Now, Lemma 17.20.12 of the Stacks Project claims that any short exact sequence $$0\to\mathscr{F}_1\to\mathscr{F}_2\to\mathscr{F}_3\to 0$$ induces exact sequences $$0\to\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F_3},\mathscr{G})\to\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F_2},\mathscr{G})\to\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{F_1},\mathscr{G})$$ and $$0\to\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{G},\mathscr{F_1})\to\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{G},\mathscr{F_2})\to\mathscr{H}\!\mathit{om}_{\mathcal{O}_X}(\mathscr{G},\mathscr{F_3})$$ Now, I can prove that the first (nontrivial) map is injective in both cases, but I don't see how to show exactness in the middle.

Answer 1

$\def\homs{\mathcal{H}om} \def\hom{\operatorname{Hom}} \def\O{\mathcal{O}} \def\F{\mathcal{F}} \def\G{\mathcal{G}} $As of current date, there is no "Lemma 17.20.12 of the Stacks Project". Maybe the numbering has changed since you asked the question five years ago. The result I think you are quoting is tag 01CO, whose proof is ommited .

As an alternative to the argument that the sheaf hom is a right adjoint and so it preserves limits (MooS pointed this out in the comments), we can give the following argument:

In the statement of 01CO, the complexes of sheaves of modules $$ \tag{1}\label{1} 0\to\homs_{\O_X}(\F,\G)\to\homs_{\O_X}(\F_1,\G)\to\homs_{\O_X}(\F_2,\G),\\ $$ $$ \tag{2}\label{2} 0\to\homs_{\O_X}(\F,\G)\to\homs_{\O_X}(\F,\G_1)\to\homs_{\O_X}(\F,\G_2), $$ are exact on sections, i.e., for each open set $U\subset X$, the sequences $$ 0\to\hom_{\O_U}(\F|_U,\G|_U)\to\hom_{\O_U}(\F_1|_U,\G|_U)\to\hom_{\O_U}(\F_2|_U,\G|_U),\\ \phantom{a}\\ 0\to\hom_{\O_U}(\F|_U,\G|_U)\to\hom_{\O_U}(\F|_U,\G_1|_U)\to\hom_{\O_U}(\F|_U,\G_2|_U), $$ are exact. This follows from the fact that $\mathsf{Mod}(\O_U)$ is an abelian category and tag 05AA. Now use the fact that taking filtered colimits of abelian groups is an exact operation (tag 00DB) to conclude that the sequences \eqref{1} and \eqref{2} are exact on stalks, and thus exact.