ScalingStacks

Proof. Tensor products and pullbacks always preserve colimits, and in our setting $f_{*}$ is colimit preserving as well. Therefore for any $M$ the functors $f_{*}M\otimes(-)$ and $f_{*}(M\otimes f^{*}-)$ define colimit preserving endofunctors of $\Mod_{A}$. Hence both functors are determined by their value on $A$, and canonically take the value $f_{*}M$. We find that the natural map is an equivalence. ∎