Lemma 9.4.6.24. Let $\kappa < \lambda \leq \mu $ be uncountable regular cardinals, let $\operatorname{\mathcal{K}}$ be an $\infty $-category which is $\mu $-small and $\lambda $-filtered, and suppose we are given a diagram
\[ \mathscr {F}: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }. \]
Assume that, for each $K \in \operatorname{\mathcal{K}}$, the Kan complex $\mathscr {F}(K)$ is essentially $\kappa $-small. Then the colimit $\varinjlim (\mathscr {F} )$ is also essentially $\kappa $-small.
Proof.
Enlarging $\mu $ if necessary, we may assume that $\lambda \trianglelefteq \mu $. In this case, Theorem 9.1.8.7 guarantees the existence of a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{K}}$, where $(A, \leq )$ is a $\mu $-small $\lambda $-directed partially ordered set. By virtue of Corollary 5.6.5.18, we may assume that $\mathscr {F}$ is obtained from (strictly) commutative diagram of Kan complexes
\[ (A, \leq ) \rightarrow \operatorname{Kan}\quad \quad (\alpha \in A) \mapsto X_{\alpha }. \]
In this case, we can use Variant 9.1.6.4 to identify the colimit $\varinjlim (\mathscr {F})$ (formed in the $\infty $-category $\operatorname{\mathcal{S}}_{< \mu }$) with the colimit $X = \varinjlim _{\alpha } X_{\alpha }$ (formed in the ordinary category of Kan complexes). We will complete the proof by showing that $X$ is essentially $\kappa $-small.
It follows from Lemma 9.4.6.23 that the set of connected components $\pi _0(X)$ is $\kappa $-small. By virtue of Proposition 4.9.6.1, it will suffice to show that for every vertex $x \in X$ and every integer $n > 0$, the homotopy group $G = \pi _{n}(X,x)$ is $\kappa $-small. Choose an index $\alpha \in A$ such that $x$ lifts to a vertex $x_{\alpha } \in X_{\alpha }$. For each $\beta \geq \alpha $, let $x_{\beta }$ denote the image of $x_{\alpha }$ in the Kan complex $X_{\beta }$. Then $G$ can be identified with the $\lambda $-directed colimit $\varinjlim _{\beta \geq \alpha } \pi _{n}( X_{\beta }, x_{\beta } )$, which is $\kappa $-small by virtue of Lemma 9.4.6.23.
$\square$