Suppose that $(X, \mathcal{O}_X)$ is a ring space. If $\mathcal{F}, \mathcal{G}$ are sheaves of $\mathcal{O}_X$-modules then the assignment $$U \mapsto \text{Hom}_{\mathcal{O}_X|U}(\mathcal{F}|U, \mathcal{G}|U)$$ makes a sheaf of $\mathcal{O}_X$-modules. (The restriction should be the natural one.)
The question is to show that the functor $\text{Hom}_{\mathcal{O}_X}(\cdot, \mathcal{G})$ is left-exact (for fixed sheaf $\mathcal{G}$) i.e. if we have an exact sequence of sheaves $$\mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0$$ then the sequence $$0 \rightarrow \text{Hom}_{\mathcal{O}_X}(\mathcal{F}'', \mathcal{G}) \rightarrow \text{Hom}_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) \rightarrow \text{Hom}_{\mathcal{O}_X}(\mathcal{F}', \mathcal{G})$$ is exact.
My attempt is as follow: by definition of exact-sequence-of-sheaves, we have to show that the induced sequence on stalks $$0 \rightarrow (\text{Hom}_{\mathcal{O}_X}(\mathcal{F}'', \mathcal{G}))_x \rightarrow (\text{Hom}_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}))_x \rightarrow (\text{Hom}_{\mathcal{O}_X}(\mathcal{F}', \mathcal{G}))_x$$ is exact. Let's consider the first one where we need to show the map is injective. Let's take two germs $f, g \in (\text{Hom}_{\mathcal{O}_X}(\mathcal{F}'', \mathcal{G}))_x$ such that their images are equals in $(\text{Hom}_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}))_x$. Let denote the map $ \mathcal{F} \rightarrow \mathcal{F}''$ by $\delta$. Then we by definition of the stalk (by direct limit), we have $$(f \circ \delta)_W = (g \circ \delta)_W \text{ i.e. } f_W \circ \delta_W = g_W \circ \delta_W$$ (as sheaf hom) for some open set $W$ containing $x$; and we want to show that $f_Z = g_Z$ for some open set $Z$ containing $x$. But this does not seems possible since we only know that $\delta_W$ is surjective under the limit. So I don't find anyway to produce such set $Z$.
EDIT: Lemma 16.3 in this document is probably what I need. But unfortunately, the proof is omitted. Alternatively, it was also stated without proof that
whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks
on Wikipedia.