$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.1.4.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $K$ be a simplicial set. Then:
- $(1)$
A morphism $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if the composition $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$.
- $(2)$
A morphism $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if the composition $F \circ \overline{u}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.
In particular, the equivalence $F$ preserves $K$-indexed limits and colimits.
Proof.
We will prove $(1)$; the proof of $(2)$ is similar. Let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a diagram and set $u = \overline{u}|_{K}$. We then have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{ / \overline{u} } \ar [r] \ar [d] & \operatorname{\mathcal{D}}_{ / (F \circ \overline{u} )} \ar [d] \\ \operatorname{\mathcal{C}}_{/u} \ar [r] & \operatorname{\mathcal{D}}_{ / (F \circ u )}. } \]
Since $F$ is an equivalence of $\infty $-categories, the horizontal maps in this diagram are also equivalences of $\infty $-categories (Corollary 4.6.4.19). It follows that the left vertical map is an equivalence of $\infty $-categories if and only if the right vertical map is an equivalence of $\infty $-categories. The desired result now follows from the criterion of Proposition 7.1.3.12.
$\square$