Given a ringed space (or more generally, a ringed site) $(X,\mathcal O_X)$, we can define internal Hom in the category of $\mathcal O_X$-modules ($\mathcal{Hom}(\mathcal F,\mathcal G)$ is defined as $U\mapsto\operatorname{Hom}_{\mathcal O_X}(\mathcal F\vert_U,\mathcal G\vert_U)$), and that the functor $\mathcal{Hom}(\mathcal F,-)$ admits a left adjoint $-\otimes_{\mathcal O_X}\mathcal F$.
I wonder whether we can put this into a more general setting. We need to notice that, for each open $U\subseteq X$, we have a category associated to $U$, namely the category $M_U$ of $\mathcal O_X(U)$-modules (in order to avoid set-theoretic issues, one can assume $M_U$ is small). In each $M_U$, we have internal Hom and tensor product. Can we view the sheaf-Hom and sheaf tensor products as the result of gluing these functors on the open sets?
We'd note that the proof of the adjointness of sheaf Hom and tensor products is quite formal, not really related to constructions of $\operatorname{Hom}$ and $\otimes$, or any commutative algebra, but only their categorical properties, which hints me that one might do it more conceptually.