Lemma 6.4 (0NMV) · ScalingStacks

Lemma 6.4. Let ${\mathcal{D}}$ be a pointed presentable $\infty$ -category. Then, we have an equivalence of $\infty$ -categories

\mathrm{Fun}^{\mathrm{L}}(\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*},{\mathcal{D}})\simeq\mathrm{Fun}_{\mathrm{flt}}(\Cat_{\infty}^{\perf},{\mathcal{D}})\,,

where the right-hand side denotes the $\infty$ -category of morphisms of $\infty$ -categories which preserve filtered colimits.

Proof. The proof is a consequence of the equivalences

	$\displaystyle\mathrm{Fun}^{\mathrm{L}}(\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*},{\mathcal{D}})$	$\displaystyle\simeq\mathrm{Fun}((\Cat_{\infty}^{\perf})^{\omega},{\mathcal{D}})$
		$\displaystyle\simeq\mathrm{Fun}_{\mathrm{flt}}(\Ind((\Cat_{\infty}^{\perf})^{\omega}),{\mathcal{D}})$
		$\displaystyle\simeq\mathrm{Fun}_{\mathrm{flt}}(\Cat_{\infty}^{\perf},{\mathcal{D}})\,,$

where the first follows from [52, 5.1.5.6] and the fact that ${\mathcal{D}}$ is pointed, and the last follows from corollary 4.25. ∎