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Corollary 7.3.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, let $F_0: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

  • The functor $F_0$ admits a left Kan extension $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the diagram

    \[ \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}^{0} \xrightarrow {F_0} \operatorname{\mathcal{D}} \]

    has a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$.

  • The functor $F_0$ admits a right Kan extension $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the diagram

    \[ \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{C}}^{0} \xrightarrow {F_0} \operatorname{\mathcal{D}} \]

    has a limit in the $\infty $-category $\operatorname{\mathcal{D}}$.

Proof. The first assertion follows by applying the criterion of Proposition 7.3.5.5 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$, and the second assertion follows by a similar argument. $\square$