Proposition 7.6.2.15 (Covariant Transport as a Kan Extension). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories having essentially small fibers, let $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$ denote the constant functor $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$ taking the value $\Delta ^0$, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be any functor. Suppose we are given a natural transformation $\beta : \underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F} \circ U$. The following conditions are equivalent:
- $(1)$
The natural transformation $\beta $ exhibits $\mathscr {F}$ as a left Kan extension of $\underline{ \Delta ^0}_{\operatorname{\mathcal{E}}}$ along $U$ (in the sense of Variant 7.3.1.5).
- $(2)$
The commutative diagram
7.55\begin{equation} \begin{gathered}\label{equation:covariant-transport-as-Kan-extension} \xymatrix { \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^{\beta } & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}\ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^{ \mathscr {F} } & \operatorname{\mathcal{S}}} \end{gathered} \end{equation}is a categorical pullback square.