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

Proposition Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, let $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be another functor having restrictions $F_0 = F|_{ \operatorname{\mathcal{K}}}$ and $F_{1} = F|_{\operatorname{\mathcal{C}}}$, so that the composition

\[ \Delta ^1 \times \operatorname{\mathcal{K}}\simeq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]

determines a natural transformation $\beta : F_0 \rightarrow F_1 \circ \delta $. The following conditions are equivalent:


The functor $F$ is left Kan extended from the full subcategory $\operatorname{\mathcal{K}}\subseteq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ (in the sense of Definition


The natural transformation $\beta $ exhibits $F_1$ as a left Kan extension of $F_0$ along $\delta $ (in the sense of Variant

Proof. By virtue of Exercise, it will suffice to show that for every object $C \in \operatorname{\mathcal{C}}$, the following conditions are equivalent:

$(1_ C)$

The functor $F$ is left Kan extended from $\operatorname{\mathcal{K}}$ at $C$ (in the sense of Definition

$(2_ C)$

The natural transformation $\beta $ satisfies condition $(\ast _ C)$ of Variant

For the remainder of the proof, let us regard the object $C \in \operatorname{\mathcal{C}}$ as fixed, and set $\operatorname{\mathcal{K}}_{/C} = \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Let $\pi : \Delta ^2 \times \operatorname{\mathcal{K}}_{/C} \rightarrow ( \Delta ^1 \times \operatorname{\mathcal{K}}_{/C} )^{\triangleright }$ be the functor which is the identity on $\Delta ^1 \times \operatorname{\mathcal{K}}_{/C}$ and carries $\{ 2\} \times \operatorname{\mathcal{K}}_{/C}$ to the cone point of $( \Delta ^1 \times \operatorname{\mathcal{K}}_{/C} )^{\triangleright }$. Let $\sigma $ denote the composite map

\begin{eqnarray*} \Delta ^2 \times \operatorname{\mathcal{K}}_{/C} & \xrightarrow {\pi } & ( \Delta ^1 \times \operatorname{\mathcal{K}}_{/C} )^{\triangleright } \\ & \simeq & (( \operatorname{\mathcal{K}}\times \Delta ^1) \times _{ \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}} ( \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})_{/C})^{\triangleright } \\ & \rightarrow & \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\\ & \xrightarrow {F} & \operatorname{\mathcal{D}}. \end{eqnarray*}

We will regard $\sigma $ as a $2$-simplex in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{K}}_{/C}, \operatorname{\mathcal{D}})$, which we display as a diagram

\[ \xymatrix@R =50pt@C=50pt{ & (F_{1} \circ \delta )|_{ \operatorname{\mathcal{K}}_{/C} } \ar [dr] & \\ F_0|_{ \operatorname{\mathcal{K}}_{/C} } \ar [ur] \ar [rr] & & \underline{ F_1(C) } } \]

which witnesses the bottom horizontal map as the natural transformation $\beta _{C}$ appearing in condition $(\ast _ C)$. By construction, this natural transformation $\beta _{C}$ is given by the composite map

\[ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \times \operatorname{\mathcal{K}}_{/C} \rightarrow (\operatorname{\mathcal{K}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}, \]

so the equivalence $(1_ C) \Leftrightarrow (2_ C)$ is a special case of Remark $\square$