Let $R$ be a ring and $X,Y$ two $R$-schemes, which you may assume to be noetherian or anything reasonable you like. Is it possible to "construct" $\text{Qcoh}(X \times_R Y)$ out of $\text{Qcoh}(X)$ and $\text{Qcoh}(X)$ in the $2$-category of all cocomplete $R$-linear tensor categories?

For example if $X,Y$ are affine, then $\text{Qcoh}(X \times_R Y)$ is the $2$-coproduct of $\text{Qcoh}(X)$ and $\text{Qcoh}(Y)$. I think I can prove it also when $X,Y$ are projective over $R$.