Kerodon

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

Corollary 4.6.4.18. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty$-categories, and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$. Then the induced functors

$G': \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{/(G \circ F)} \quad \quad G'': \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{D}}_{(G \circ F)/}$

are equivalences of $\infty$-categories.

Proof. We will show that $G'$ is an equivalence of $\infty$-categories; the analogous statement for $G''$ follows by a similar argument. Note that we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/F} \ar [r]^-{G'} \ar [d] & \operatorname{\mathcal{D}}_{ / (G \circ F) } \ar [d] \\ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \ar [r]^-{\overline{G}'} & \operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \{ G \circ F \} , }$

where the vertical maps are equivalences of $\infty$-categories by virtue of Theorem 4.6.4.16. It will therefore suffice to show that $\overline{G}'$ is an equivalence of $\infty$-categories. This follows by applying Corollary 4.5.2.25 to the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\Delta ^0 \diamond K, \operatorname{\mathcal{C}}) \ar [r]^-{G \circ } \ar [d] & \operatorname{Fun}(\Delta ^0 \diamond K, \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \ar [r]^-{G \circ } & \operatorname{Fun}(K,\operatorname{\mathcal{D}}); }$

here the vertical maps are isofibrations by virtue of Corollary 4.4.5.3, and the horizontal maps are equivalences by virtue of Remark 4.5.1.16. $\square$