The reason that I ask this is because I have recently defined the tensor product of $O_X$ modules $\mathscr{F}, \mathscr{G}$ (where $X$ is a ringed space) by defining the presheaf at each open set $U \subset X$ as $\mathscr{F}(U) \otimes_{O_X(U)} \mathscr{G}(U)$ and then sheafifying to obtain the tensor product in the category of $O_X$ modules.
However, I have read in many places that, like the module tensor product, this tensor product also characterizes bilinear maps $\mathscr{F} \times \mathscr{G} \to \mathscr{H}$ of $O_X$ modules. However, it is not entirely clear to me what this definition should be, and it seems that every place I look explicitly states that they won't define what it means.
This would simply be a morphism $\phi$ of sheaves of sets (which in general is almost certainly not $\mathscr{O}_X$-linear, nor even a morphism of sheaves of abelian groups) such that for any $U \subseteq X$ open, $\phi_U : \mathscr{F}(U) \times \mathscr{G}(U) \to \mathscr{H}(U)$ is $\mathscr{O}_X(U)$-bilinear, i.e. for any $x_1, x_2 \in \mathscr{F}(U)$, $y_1, y_2 \in \mathscr{G}(U)$, $a \in \mathscr{O}_X(U)$, we have: $$\phi_U(x_1 + x_2, y_1) = \phi_U(x_1, y_1) + \phi_U(x_2, y_1); \\ \phi_U(x_1, y_1 + y_2) = \phi_U(x_1, y_1) + \phi_U(x_1, y_2); \\ \phi_U(a x_1, y_1) = \phi_U(x_1, a y_1) = a \phi_U(x_1, y_1).$$