Proposition 7.3.8.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose we are given a right fibration of $\infty $-categories $V: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{B}}^{0} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{B}}$. Then, for every object $B \in \operatorname{\mathcal{B}}$, the functor $F \circ V$ is $U$-left Kan extended from $\operatorname{\mathcal{B}}^{0}$ at $B$ if and only if $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $V(B)$.
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Proof. Set $C = V(B)$, and let $F_{C}$ denote the composite map
\[ (\operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C})^{\triangleright } \rightarrow (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}. \]
We wish to show that $F_{C}$ is a $U$-colimit diagram if the composite map
\[ (\operatorname{\mathcal{B}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}_{/B})^{\triangleright } \rightarrow (\operatorname{\mathcal{B}}_{/B})^{\triangleright } \rightarrow \operatorname{\mathcal{B}}\xrightarrow {V} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]
is a $U$-colimit diagram. By virtue of Corollary 7.2.2.2, it will suffice to show that the natural map
\[ \theta : \operatorname{\mathcal{B}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}_{/B} \rightarrow \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \]
is right cofinal. By construction, $\theta $ is a pullback of the map $V_{/B}: \operatorname{\mathcal{B}}_{/B} \rightarrow \operatorname{\mathcal{C}}_{ / V(B) }$. Our assumption that $V$ is a right fibration guarantees that $V_{/B}$ is a trivial Kan fibration (Corollary 4.3.7.13). It follows that $\theta $ is also a trivial Kan fibration, and therefore right cofinal by virtue of Corollary 7.2.1.13. $\square$