Omitting some technicalities, the base-change theorem for quasicoherent sheaves says that if we have the following diagram of (derived) schemes $\require{AMScd}$ \begin{CD} X \times_S Y @>\pi_2>> Y\\ @V \pi_1 V V @VV p_Y V\\ X @>>p_X> S \end{CD} where the fibre product $X \times_S Y$ is derived, then there is an isomorphism $$p_Y^*p_{X,*}(F) \cong \pi_{2,*}\pi_1^*(F)$$ where all the functors are derived and F is a (derived) quasicoherent sheaf.
This result implies a commutative diagram of stable $\infty$-categories, namely, $\require{AMScd}$ \begin{CD} \mathrm{QCoh}(X) @>p_{X,*}>> \mathrm{QCoh}(S)\\ @V \pi_1^* V V @VV p_Y^* V\\ \mathrm{QCoh}(X) \otimes_{\mathrm{QCoh}(S)} \mathrm{QCoh}(Y) @>>\pi_{2,*}> \mathrm{QCoh}(Y) \end{CD}
My question is about a purely categorical generalization of the above. Namely, is it correct to say something like whenever I have two stable presentable categories $\mathscr{A}$ and $\mathscr{B}$ which are modules over some symmetric monoidal category $\mathscr{S}$, together with left adjoints to the functors $p_{A,*}: \mathscr{A} \to \mathscr{S}$ and $p_{B,*}: \mathscr{B} \to \mathscr{S}$, that we would have a similar commutative diagram looking like $\require{AMScd}$ \begin{CD} \mathscr{A} @>p_{A,*}>> \mathscr{S}\\ @V \pi_1^* V V @VV p_B^* V\\ \mathscr{A} \otimes_{\mathscr{S}} \mathscr{B} @>>\pi_{2,*}> \mathscr{B} \end{CD}
I feel like such a statement should exist, and I'm hoping it is written somewhere in the literature or is easy to prove for an expert. Thanks in advance.
