Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.2.4.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then $\operatorname{\mathcal{C}}$ is filtered if and only if $\operatorname{\mathcal{D}}$ is filtered.

Proof. By virtue of Proposition 7.2.4.10, it will suffice to show that for every (finite) simplicial set $K$, the diagonal map $\delta _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is right cofinal if and only if the diagonal map $\delta _{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{D}})$ is right cofinal. This follows by applying Corollary 7.2.1.21 to the commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{\delta _{\operatorname{\mathcal{C}}} } \ar [d]^{F} & \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \ar [d]^{ F \circ } \\ \operatorname{\mathcal{D}}\ar [r]^-{ \delta _{\operatorname{\mathcal{D}}} } & \operatorname{Fun}(K, \operatorname{\mathcal{D}}). } \]
$\square$