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Corollary 7.3.6.16. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Let $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which are left Kan extended from $\operatorname{\mathcal{C}}^{0}$, and let $\operatorname{Fun}'( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}})$ spanned by those functors $F_0$ which satisfy the following condition:

$(\ast )$

For every object $C \in \operatorname{\mathcal{C}}$, the diagram

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

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

Then the restriction map $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}})$ is a trivial Kan fibration of simplicial sets.

Proof. Apply Theorem 7.3.6.15 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$