Let $X$ and $Y$ be qcqs schemes over a field $k$ (or I am happy to assume any nice condition up to smooth (quasi-)projective varieties if that makes the folloiwng question true). Let us define $Perf(X)$ to be a dg enhancement of the category of perfect complexes on $X$. Is it true that $$ Hom_{Perf(X\times Y)}(F \boxtimes G, F' \boxtimes G') \cong Hom_{Perf(X)}(F,F')\otimes_k Hom_{Perf(Y)}(G,G') $$
as dg modules? The case of Ext groups of sheaves is discussed here Künneth formula for Ext groups. Also I think Proposition 7.4.2. in this note https://people.math.rochester.edu/faculty/doug/otherpapers/gaitsgory2.pdf together with Theorem 1.2 (1) in this paper https://arxiv.org/pdf/0805.0157v5.pdf could possibly give what I want but I want to make sure it actually does before starting to read since I am not familar with languages they are using. I would also appreciate it if we can show the statement with language of dg categories and schemes. Thank you in advance.

Let $X$ and $Y$ be qcqs schemes over a field $k$ (or I am happy to assume any nice condition up to smooth (quasi-)projective varieties if that makes the folloiwng question true). Let us define $Perf(X)$ to be a dg enhancement of the category of perfect complexes on $X$. Is it true that $$ Hom_{Perf(X\times Y)}(F \boxtimes G, F' \boxtimes G') \cong Hom_{Perf(X)}(F,F')\otimes_k Hom_{Perf(Y)}(G,G') $$
as dg modules? The case of Ext groups (non-dg case) is discussed here Künneth formula for Ext groups. Also I think Proposition 7.4.2. in this note https://people.math.rochester.edu/faculty/doug/otherpapers/gaitsgory2.pdf together with Theorem 1.2 (1) in this paper https://arxiv.org/pdf/0805.0157v5.pdf could possibly give what I want but I want to make sure it actually does before starting to read since I am not familar with languages they are using. I would also appreciate it if we can show the statement with language of dg categories and schemes. Thank you in advance.