Proposition 9.2.8.4. Let $X$ be a Kan complex. The following conditions are equivalent:
The Kan complex $X$ is essentially finite in the sense of Definition 9.2.7.1: that is, there exists a finite simplicial set $K$ and a weak homotopy equivalence $f: K \rightarrow X$.
The Kan complex $X$ is essentially finite when viewed as an $\infty $-category, in the sense of Definition 9.2.8.1: that is, there exists a finite simplicial set $K$ and a categorical equivalence $K \rightarrow X$.
Proof. The implication $(2) \Rightarrow (1)$ is trivial, since every categorical equivalence of simplicial sets is a weak homotopy equivalence (Remark 4.5.3.4). We now prove the converse. Assume that there exists a weak homotopy equivalence $f: K \rightarrow X$, where $K$ is a finite simplicial set. Then $f$ factors as a composition $K \xrightarrow {i} \operatorname{\mathcal{C}}\xrightarrow {F} X$, where $\operatorname{\mathcal{C}}$ is an $\infty $-category and $i$ is a categorical equivalence of simplicial sets (Proposition 4.1.3.2). Let $W$ be the collection of all edges of $K$. Since $f$ is a weak homotopy equivalence, it exhibits the Kan complex $X$ as a localization of $K$ with respect to $W$ (Proposition 6.3.1.21). It follows that the functor $F$ exhibits $X$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $i(W)$. Since $\operatorname{\mathcal{C}}$ is an essentially finite $\infty $-category, Proposition 9.2.8.3 implies that $X$ is also an essentially finite $\infty $-category. $\square$