Corollary 7.3.6.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{D}}$ admits colimits indexed by the simplicial set $K_{/C} = K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Then the restriction functor
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow { \circ \delta } \operatorname{Fun}(K, \operatorname{\mathcal{D}}) \]
has a left adjoint, which carries each diagram $F_0: K \rightarrow \operatorname{\mathcal{D}}$ to a left Kan extension of $F_0$ along $\delta $.