($\OO_X$-modules)
An \(\OO_X\)-module is a sheaf \(\mcf\) on \(X\) with each \(\mcf(U)\) an \(\OO_X(U)\)-module, compatibly with restriction: \(\ro{(rm)}{V} = \ro{r}{V}\ro{m}{V}\). The tensor product \(\mcf \tensor_{\OO_X} \mcg\) is the sheafification of \(U \mapsto \mcf(U) \tensor_{\OO_X(U)} \mcg(U)\). The sheaf hom \(\sheafhom_{\OO_X}(\mcf, \mcg)\) is \(U \mapsto \Hom_{\ro{\OO_X}{U}}(\ro{\mcf}{U}, \ro{\mcg}{U})\). A sheaf of ideals is a subsheaf \(\mci \subseteq \OO_X\) of \(\OO_X\)-modules.
The presheaf \(U \mapsto \mcf(U) \tensor_{\OO_X(U)} \mcg(U)\) need not be a sheaf: on \(X=\PP^1_k\), its value on \(X\) for \(\mcf=\OO(1)\) and \(\mcg=\OO(-1)\) is \(H^0(\OO(1))\tensor_kH^0(\OO(-1))=0\), while \(\OO(1)\tensor\OO(-1)\cong\OO_X\) has the global section \(1\). Morphisms of sheaves defined on the members of an open cover and agreeing on overlaps glue uniquely, so \(U\mapsto\Hom_{\ro{\OO_X}{U}}(\ro{\mcf}{U}, \ro{\mcg}{U})\) is already a sheaf.
The functor \(\wait\tensor_{\OO_X}\mcg\) is right exact, and \(\sheafhom_{\OO_X}(\mcf,\wait)\) is left exact. For every \(x\in X\), \((\mcf \tensor \mcg)_x \cong \mcf_x \tensor_{\OO_{X,x}} \mcg_x\). If \(X\) is Noetherian and \(\mcf\) is coherent, then \(\sheafhom(\mcf,\mcg)_x \cong \Hom_{\OO_{X,x}}(\mcf_x, \mcg_x)\) [@Har10a, Proposition III.6.8].
\(\mods{\OO_X}\) has enough injectives [@Har10a, Proposition III.2.2], but it need not have enough projectives. On \(\PP_k^1\) over an infinite field, no projective object surjects onto \(\OO_X\), as proved in P-AGH362NOPROJECTIVES. In contrast, on a one-point ringed space with structure ring \(A\), a sheaf of modules is just an \(A\)-module; free-module surjections show that this category has enough projectives.