In Toen's paper The homotopy theory of dg-algebras and derived Morita theory, Theorem 8.15, he essentially proved the following result.
Let $X$ and $Y$ be two smooth and proper schemes over $k$. Let $\mathcal{E}$ be an object in $D_{qcoh}(X\times Y)$ such that the integral transform $$ \Phi_{\mathcal{E}}:=Rp_{Y,*}(p^*_X(-)\otimes^{L}\mathcal{E}) $$ maps $D_{perf}(X)$ to $D_{perf}(Y)$. Then $\mathcal{E}$ itself is a perfect complex on $X\times Y$.
I wonder if this result is due to Toen or it already exists in previous works (by himself or other authors.)
