$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.6.4.19. 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 4.6.4.17. It will therefore suffice to show that $\overline{G}'$ is an equivalence of $\infty $-categories, which is a special case of Remark 4.6.4.4.
$\square$