ScalingStacks

Let $B,B^{\prime}$ be two $k$-algebras, projective as $k$-modules. Let $L\in\operatorname{Ho}\nolimits^{b}\bigl((A\otimes B^{\operatorname{opp}\nolimits})\operatorname{\!-Mod}\nolimits\bigr)$ and $M\in\operatorname{Ho}\nolimits^{b}((B\otimes B^{\prime{\operatorname{opp}\nolimits}})\operatorname{\!-Mod}\nolimits)$. Assume the components of $M$ are projective right $B^{\prime}$-modules. We deduce from (1) an isomorphism

(2)

$$\operatorname{Tot}\nolimits^{13\to 1,2\to 2}\bigl(C_{A\otimes B^{\operatorname{opp}\nolimits}}(L)\otimes_{B}M\bigr)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C_{A\otimes B^{\prime{\operatorname{opp}\nolimits}}}\bigl(\operatorname{Tot}\nolimits(L\otimes_{B}M)\bigr).$$