$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.3.3.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty $-categories, and let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a coreflective subcategory of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
- $(2)$
Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$ which exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$ (Definition 6.2.2.1). Then $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.
- $(3)$
Let $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a right adjoint to the inclusion. If $e$ is a morphism in $\operatorname{\mathcal{C}}$ and $T(e)$ is an isomorphism in $\operatorname{\mathcal{C}}_0$, then $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.
Proof.
Let $Y$ be an object of $\operatorname{\mathcal{C}}$. By assumption, there exists an object $X \in \operatorname{\mathcal{C}}^{0}$ and a morphism $e: X \rightarrow Y$ which exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$. Then $e$ is final when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$. It follows that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $Y$ if and only if $F(e )$ is $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$; in particular, this condition is independent of the choice of $e$. Allowing the object $Y$ to vary, we deduce the equivalence $(1) \Leftrightarrow (2)$.
Using Lemma 6.2.2.16, we can choose a functor $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{0}$ and a natural transformation $\epsilon : T \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ which exhibits $T$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection functor, so that $T$ is right adjoint to the inclusion of $\operatorname{\mathcal{C}}^{0}$ into $\operatorname{\mathcal{C}}$ (Proposition 6.2.2.17). Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. If $e$ exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$, then $T(e)$ is an isomorphism in $\operatorname{\mathcal{C}}^{0}$, which shows immediately that $(3)$ implies $(2)$. Conversely, suppose that $(2)$ is satisfied and that $T(e)$ is an isomorphism in $\operatorname{\mathcal{C}}^{0}$. We then have a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ (F \circ T)(X) \ar [r]^-{ (F \circ T)(e) } \ar [d]^{ F(\epsilon _{X})} & (F \circ T)(Y) \ar [d]^{ F( \epsilon _ Y) } \\ F(X) \ar [r]^-{ F(e) } & F(Y) } \]
in the $\infty $-category $\operatorname{\mathcal{D}}$, where the upper horizontal map is an isomorphism and the vertical maps are $U$-cocartesian. Using Corollary 5.1.2.4, we see that $F(e)$ is also $U$-cocartesian.
$\square$