Remark 9.2.8.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Choose a weak homotopy equivalence $F: \operatorname{\mathcal{C}}\rightarrow X$, where $X$ is a Kan complex (for example, we could take $X = \operatorname{Ex}^{\infty }(\operatorname{\mathcal{C}})$). Suppose that $\operatorname{\mathcal{C}}$ is essentially finite: that is, that there exists a categorical equivalence of simplicial sets $G: K \rightarrow \operatorname{\mathcal{C}}$. Then the composite map $(G \circ F): K \rightarrow X$ is a weak homotopy equivalence, so that $X$ is an essentially finite when regarded as a Kan complex. It follows from Proposition 9.2.8.4 that $X$ is also essentially finite when regarded as an $\infty $-category.
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