ScalingStacks

Proof. For any small stable idempotent-complete $\infty$-category ${\mathcal{B}}$, we can consider the functor

$$(-)\widehat{\otimes}{\mathcal{B}}\colon\Cat_{\infty}^{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}.$$

By lemma 9.35, the composed morphism

$$\Cat_{\infty}^{\perf}\stackrel{{\scriptstyle(-)\widehat{\otimes}{\mathcal{B}}}}{{\longrightarrow}}\Cat_{\infty}^{\perf}\stackrel{{\scriptstyle{\mathcal{U}}_{\mathrm{loc}}}}{{\longrightarrow}}{\mathcal{M}}_{\mathrm{loc}}$$

is a localizing invariant. Thus, we obtain a commutative diagram

with $\overline{(-)\widehat{\otimes}{\mathcal{B}}}$ a colimit-preserving functor such that

$$\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}})\widehat{\otimes}{\mathcal{B}}}\simeq{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}})\,.$$

Now, recall from theorem 3.7 that since ${\mathcal{B}}$ is smooth and proper, it is also dualizable (in the symmetric monoidal $\infty$-category $\Cat_{\infty}^{\perf}$ of idempotent-complete small stable $\infty$-categories). Therefore, we have an adjunction (on the left) [53, 4.2.5.6], which induces an adjunction (on the right)

with $\overline{(-)\widehat{\otimes}{\mathcal{B}}^{\op}}$ a colimit preserving morphism, such that

$$\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}})\widehat{\otimes}{\mathcal{B}}^{\op}}\simeq{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}\otimes{\mathcal{B}}^{\op})\,.$$

The proof now follows from the following equivalences of spectra

	$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{B}}),{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))$	$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega})\widehat{\otimes}{\mathcal{B}}},{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}))$
		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}})\widehat{\otimes}{\mathcal{B}}^{\op}})$
		$\displaystyle\simeq$	$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{loc}}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op}))$
		$\displaystyle\simeq$	$\displaystyle I\mspace{-6.mu}K({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op})$
		$\displaystyle\simeq$	$\displaystyle I\mspace{-6.mu}K(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))\,.$

Finally, since in the adjunction

the morphism $\overline{(-)\widehat{\otimes}{\mathcal{B}}^{\op}}$ preserves colimits, the object ${\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega})$ is compact, and $\overline{{\mathcal{U}}_{\mathrm{loc}}({\mathcal{S}}_{\infty}^{\omega})\otimes{\mathcal{B}}}\simeq{\mathcal{U}}_{\mathrm{loc}}({\mathcal{B}})$, we conclude that ${\mathcal{U}}_{\mathrm{loc}}({\mathcal{B}})$ is compact. ∎