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Variant (Left Kan Extensions). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose we are given a simplicial set $K$ together with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta : F_0 \rightarrow F \circ \delta $, as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {<=}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\beta } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}. } \]

We will say that $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ if, for every object $C \in \operatorname{\mathcal{C}}$, the following condition is satisfied:

$(\ast _ C)$

Let $\beta _{C}$ denote a composition of the natural transformations

\[ F_0|_{ K_{/C} } \xrightarrow {\beta } (F \circ \delta )|_{ K_{/C} } \xrightarrow { F(\gamma ) } \underline{ F(C) } \]

(formed in the $\infty $-category $\operatorname{Fun}( K_{/C}, \operatorname{\mathcal{D}})$), where $\gamma : \delta |_{ K_{/C} } \rightarrow \underline{C}$ is defined in Notation Then $\beta _{C}$ exhibits $F(C)$ as a colimit of the diagram

\[ K_{/C} = K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}}, \]

in the sense of Definition