# Kerodon

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

Proposition 7.3.2.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $F_0$ denote the restriction of $F$ to a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, and let $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ denote the inclusion functor. Then:

• The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1) if and only if the identity transformation $\operatorname{id}: F_0 \rightarrow F \circ \iota$ exhibits $F$ as a left Kan extension of $F_0$ along $\iota$ (in the sense of Variant 7.3.1.5).

• The functor $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1) if and only if the identity transformation $\operatorname{id}_{F_0}: F \circ \iota \rightarrow F_0$ exhibits $F$ as a right Kan extension of $F_0$ along $\iota$ (in the sense of Definition 7.3.1.2).

Proof. Fix an object $C \in \operatorname{\mathcal{C}}$. It follows from Remark 7.1.2.6 that the composition

$(\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$

is a colimit diagram in $\operatorname{\mathcal{D}}$ if and only if the natural transformation $\operatorname{id}_{ F_0 }$ satisfies condition $(\ast _ C)$ of Variant 7.3.1.5. The first assertion follows by allowing the object $C$ to vary, and the second follows by a similar argument. $\square$