$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.3.6.11. Let $\overline{F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a cocartesian fibration. Suppose that $\operatorname{\mathcal{C}}$ contains an initial object $C$ having image $E = \overline{F}(C)$ and that the $\infty $-category $\operatorname{\mathcal{D}}_{E} = \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ has an initial object. Then the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ has an initial object. Moreover, an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is initial if and only if it satisfies the following conditions:
- $(1)$
The image $F(C)$ is an initial object of the $\infty $-category $\operatorname{\mathcal{D}}_{E}$.
- $(2)$
The functor $F$ carries each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.
Proof.
It follows from Corollary 7.3.6.10 that any object of $\operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ which satisfies conditions $(1)$ and $(2)$ is initial. It will therefore suffice to show that there exists an object $F \in \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $(1)$ and $(2)$ (any other initial object of $\operatorname{Fun}_{ /\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ will be isomorphic to $F$, and will therefore also satisfy $(1)$ and $(2)$).
Let $\operatorname{\mathcal{C}}^{\mathrm{init}}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by its initial objects, and let $D$ be an initial object of the $\infty $-category $\operatorname{\mathcal{D}}_{E}$. Since $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex, we can lift $\overline{F}|_{ \operatorname{\mathcal{C}}^{\mathrm{init}}}$ to a functor $F_0: \operatorname{\mathcal{C}}^{\mathrm{init}} \rightarrow \operatorname{\mathcal{D}}$ satisfying $F_0(C) = D$. Corollary 7.3.5.11 then guarantees that $F_0$ admits a $U$-left Kan extension $F \in \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, which satisfies condition $(2)$ by virtue of Corollary 7.3.3.13.
$\square$