Kerodon

Proof. Assume that $\operatorname{\mathcal{D}}$ is $\kappa $-filtered; we will show that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered and that $\iota $ is right cofinal (the reverse implication is a special case of Corollary 9.1.5.13). We first show that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered. Let $K$ be a $\kappa $-small simplicial set and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram; we wish to show that $f$ can be extended to a map $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. If $K = \emptyset $, this follows from our assumption that $\operatorname{\mathcal{C}}$ is nonempty. Otherwise, our assumption that $\operatorname{\mathcal{D}}$ is $\kappa $-filtered guarantees that we can extend $f$ to a diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$, which automatically factors through the full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$.

To complete the proof, we must show that $\iota $ is right cofinal. Using the criterion of Theorem 9.1.4.5, we are reduced to showing that for each object $C \in \operatorname{\mathcal{C}}$, the functor $\iota _{C/}: \operatorname{\mathcal{C}}_{C/} \hookrightarrow \operatorname{\mathcal{D}}_{C/}$ is weakly right cofinal. This is clear: our assumption on $\operatorname{\mathcal{C}}$ guarantees that $\iota _{C/}$ is an isomorphism. $\square$