Proposition 7.3.3.21 (02ZG): Base Change

Proposition 7.3.3.21 (Base Change). Suppose we are given a commutative diagram of $\infty $-categories

7.13

\begin{equation} \begin{gathered}\label{equation:base-change-relative-limit} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}' \ar [rr]^{H'} \ar [dr] \ar [dd]^{G} & & \operatorname{\mathcal{E}}' \ar [dl] \ar [dd] \\ & \operatorname{\mathcal{B}}' \ar [dd] & \\ \operatorname{\mathcal{D}}\ar [dr]_{ U } \ar [rr]^(.4){H} & & \operatorname{\mathcal{E}}\ar [dl]^{V} \\ & \operatorname{\mathcal{B}}& } \end{gathered} \end{equation}

where each square is a pullback and the diagonal maps are inner fibrations. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$ be a functor of $\infty $-categories and $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then:

$(1)$: If $G \circ F$ is $H$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $H'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
$(2)$: Assume that $U$ and $V$ are cartesian fibrations and that the functor $G$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{E}}$. If $F$ is $H'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $G \circ F$ is an $H$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. Use Proposition 7.1.5.19. $\square$