Let $k$ be a field, $R,S$ commutative $k$-algebras, $A,A' \in \mathsf{Mod}(R)$ and $B,B' \in \mathsf{Mod}(S)$ modules of finite presentation. If necessary, you can assume that $R,S$ are of finite type, or any other nice condition. How can we characterize those elements in $\hom_R(A,A') \otimes_k \hom_S(B,B')$ which represent an epimorphism of $R \otimes_k S$-modules $A \otimes_k B \to A' \otimes_k B'$ under the canonical isomorphism $\hom_R(A,A') \otimes_k \hom_S(B,B') \cong \hom_{R \otimes_k S}(A \otimes_k B,A' \otimes_k B')$?

For pure tensors this is easy: If $\alpha : A \to A'$ and $\beta : B \to B'$ and wlog $A',B' \neq 0$, then $\alpha \otimes \beta$ is an epimorphism iff $\alpha$ and $\beta$ are epimorphisms.

But when is, for example, $\alpha_1 \otimes \beta_1 + \alpha_2 \otimes \beta_2$ an epimorphism? A necessary condition is that $(\alpha_1,\alpha_2) : A^{\oplus 2} \to A'$ is an epimorphism, likewise $(\beta_1,\beta_2) : B^{\oplus 2} \to B'$, but of course this won't be sufficient. I would like to have a condition which takes place in $\mathsf{Mod}(R)$ and $\mathsf{Mod}(S)$ separately.

Background. Actually I'm interested in a more general situation where $R,S$ are replaced by sufficiently nice $k$-schemes $X,Y$ and $A,A'$ (resp. $B,B'$) are quasi-coherent sheaves on $X$ (resp. $Y$) of finite presentation. I want to show (in order to prove this) that if $F : \mathsf{Qcoh}(X) \to \mathcal{C}$, $G : \mathsf{Qcoh}(Y) \to \mathcal{C}$ are cocontinuous $k$-linear tensor functors, then for every epimorphism $A \boxtimes_k B \to A' \boxtimes_k B'$ also the induced morphism $F(A) \otimes G(B) \to F(A') \otimes G(B')$ is an epimorphism. I know this (quite indirectly) when $X$ is quasiprojective. For the general case I need a "separated" characterization of epimorphisms.