ScalingStacks

Remark 4.11. The equivalence of Corollary 4.10 is naturally monoidal in the following sense. For perfect stacks $X,X^{\prime},X^{\prime\prime}$ mapping to a perfect stack $Y$, there is a convolution map

$$\qc(X\times_{Y}X^{\prime})\otimes\qc(X^{\prime}\times_{Y}X^{\prime\prime})\to\qc(X\times_{Y}X^{\prime\prime})$$

given by pulling back and pushing forward with respect to the triple product $X\times_{Y}X^{\prime}\times_{Y}X^{\prime\prime}$ (see Section 5.2). On the other hand, we have a composition map

$$\Fun_{Y}(\qc(X),\qc(X^{\prime}))\otimes\Fun_{Y}(\qc(X^{\prime}),\qc(X^{\prime\prime}))\to\Fun_{Y}(\qc(X),\qc(X^{\prime\prime})),$$

and the equivalence of the theorem intertwines these composition maps.