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

Corollary Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories and let $u: K \rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, so that $G$ induces a functor of coslice $\infty $-categories $G': \operatorname{\mathcal{D}}_{/u} \rightarrow \operatorname{\mathcal{C}}_{ / (G \circ u)}$. If the functor $G$ admits a left adjoint, then the functor $G'$ also admits a left adjoint.

Proof. We will use the criterion of Corollary Fix an object $\overline{X} \in \operatorname{\mathcal{C}}_{ / (G \circ u) }$; we wish to show that the $\infty $-category

\[ \operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}_{/u} \times _{ \operatorname{\mathcal{C}}_{ / G \circ u} } ( \operatorname{\mathcal{C}}_{ / (G \circ u) } )_{\overline{X} / } \]

has an initial object. Let $X$ denote the image of $\overline{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Unwinding the definitions, we can identify $\overline{X}$ with a morphism of simplicial sets $\overline{u}: K \rightarrow (\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/})$, and $\operatorname{\mathcal{E}}$ with the slice $\infty $-category $(\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/})_{ / \overline{u} }$. Since $G$ admits a left adjoint, the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ has an initial object (Corollary Applying Corollary, we conclude that $\operatorname{\mathcal{E}}$ also has an initial object. $\square$