ScalingStacks

Proof. Set $X=X_{1}\times X_{2}$. By Proposition 3.24, we know that the external product takes compact objects to compact objects, and $\qc(X)^{c}$ is generated by external products.

Thus it suffices to verify that for $M_{i},N_{i}\in\qc(X_{i})^{c}$ we have an equivalence

$$\Hom_{X}(M_{1}\boxtimes M_{2},N_{1}\boxtimes N_{2})\simeq\Hom_{X_{1}}(M_{1},N_{1})\otimes\Hom_{X_{2}}(M_{2},N_{2}).$$

Using the fact that each $M_{i}$ is dualizable and $p_{2}$ satisfies the projection formula (since it is perfect), we calculate

	$\displaystyle\Hom_{X}(p_{1}^{*}M_{1}\otimes p_{2}^{*}M_{2},p_{1}^{*}N_{1}\otimes p_{2}^{*}N_{2})$	$\displaystyle\simeq$	$\displaystyle\Gamma(X,p_{1}^{*}M_{1}^{\vee}\otimes p_{1}^{*}N_{1}\otimes p_{2}^{*}M_{2}^{\vee}\otimes p_{2}^{*}N_{2})$
		$\displaystyle\simeq$	$\displaystyle\Gamma(X_{2},(p_{2})_{*}(p_{1}^{*}{\mathcal{H}om}_{X_{1}}(M_{1},N_{2})\otimes p_{2}^{*}{\mathcal{H}om}_{X_{2}}(M_{2},N_{2})))$
		$\displaystyle\simeq$	$\displaystyle\Gamma(X_{2},\Hom_{X_{1}}(M_{1},N_{1})\otimes{\mathcal{H}om}_{X_{2}}(M_{2},N_{2}))$
		$\displaystyle\simeq$	$\displaystyle\Hom_{X_{1}}(M_{1},N_{1})\otimes\Hom_{X_{2}}(M_{2},N_{2})$

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