In their paper "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry" the authors show that for perfect stacks $X$ and $Y$ over $k$, and their $k$-linear $\infty$-categories of unbounded chain complexes of modules $QC(X)$ and $QC(Y)$ there are equivalences of categories $$ QC(X \times\_k Y) \simeq QC(X)\otimes\_k QC(Y)$$ and $$QC(X \times\_k Y) \simeq Fun^L\_k(QC(X),QC(Y)).$$


My question is the following: How can this be done in the absence of the underlying geometric spaces ($X$ and $Y$)? Given two perfect $k$-linear $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$ is there a natural integral transform $$\mathcal{A}\otimes\_k \mathcal{B} \to Fun\_k^L(\mathcal{A},\mathcal{B})?$$ Under what conditions on $\mathcal{A}$ and $\mathcal{B}$ is this map an equivalence?


Any reference will be very helpful.



Integral transform on noncommutative spaces
Let $DGCat\_k$ denote the $\infty$-category obtained by localizing the category of dg-categories at Morita equivalences. This is presented by the Morita model structure on the category of dg-categories. Recall that the fibrant objects are the idempotent complete dg-categories, or equivalently dg-categories for which the canonical functor $T \hookrightarrow D^c(T)$, into the dg-category of compact dg-modules, is an equivalence. Its dualizable objects are precisely the *saturated* dg-categories ([Cisinski-Tabuada](http://www.math.univ-toulouse.fr/~dcisinsk/MonoidalMotives5.pdf), Theorem 5.8). These correspond to schemes over $k$ which are smooth and proper.


Let $A$ and $B$ be small dg-categories and consider To\"en's formula
 $$Fun\_k(A,B) = D\_{rqr}(A^{op} \otimes B)$$
for the internal hom in the $\infty$-category of dg-categories, where $D\_{rqr}$ is the dg-category of right quasi-representable bimodules.
If $A$ and $B$ are saturated, this reduces to an equivalence of dg-categories
 $$Fun\_k(A,B) = D\_c(A^{op} \otimes B)$$
where $D\_c$ is the dg-category of compact dg-modules. If $A$ and $B$ are idempotent complete, and $A$ has the property of self-duality, then this gives an equivalence of dg-categories
 $$ Fun\_k(A, B) = A \otimes B. $$
It is clear that this recovers the integral transform formula for perfect complexes, in the case of smooth proper $k$-schemes.


This is only the small version of your question. In the presentable case all I can say at the moment is the following. If $A$ and $B$ are locally presentable dg-categories, compactly generated with $A\_c$ and $B\_c$ the full subcategories of compact objects, then
 $$ Fun^L\_k(A,B) = Fun^L\_k(\hat{A\_c}, \hat{B\_c}) = (A\_c^{op} \otimes B\_c)^{\wedge} = (A\_c \otimes B\_c)^{\wedge} $$
where the second equivalences uses Corollary 7.6 in To\"en's "Derived Morita theory". Here $(-)^\wedge$ denotes the free cocompletion, and I am assuming self-duality for $A$ again. This seems to recover the integral transform formula in the case of schemes, using $Perf(X \times\_k Y) = Perf(X) \otimes Perf(Y)$.


(It has been a while since I looked at this stuff, so let me know if I'm making a mistake somewhere.)


