ScalingStacks

Corollary 4.10. Let $X,X^{\prime}$ and $Y$ be perfect stacks with maps $X\rightarrow Y\leftarrow X^{\prime}$. Then there is a natural equivalence of $\infty$-categories

$$\qc(X\times_{Y}X^{\prime})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Fun_{Y}(\qc(X),\qc({X^{\prime}})).$$

In other words, the $\infty$-category of integral kernels is equivalent to the $\infty$-category of functors.