I've heard that given a ringed topos $(X,\mathcal{O}_X)$, the functor $Hom_{\mathcal{O}_X-\operatorname{Mod}}(\mathcal{O}_X, -)$ often fails to be exact. Is this only the case for the unenriched hom (global sections), i.e. $$Hom_{\mathcal{O}_X-\operatorname{Mod}}(\mathcal{O}_X, -):\mathcal{O}_X-\operatorname{Mod}\to \Gamma(\mathcal{O}_X,X)-\operatorname{Mod},$$ or is it also true in the case of the true Hom-sheaf, i.e. $$Hom_{\mathcal{O}_X-\operatorname{Mod}}(\mathcal{O}_X, -):\mathcal{O}_X-\operatorname{Mod}\to\mathcal{O}_X-\operatorname{Mod}?$$

Morally, it seems like the second case should be exact, if only for the reason that we've done nothing but relativize.

More generally, I've heard that free $\mathcal{O}_X$-modules can often fail to be projective. Is this another case of not considering the "enriched Hom object", but only considering its global sections?