Corollary 7.2.2.2 (02XQ)

Corollary 7.2.2.2. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a morphism of simplicial sets $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if the composite map

\[ A^{\triangleleft } \xrightarrow {e^{\triangleleft }} B^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]

is a $U$-limit diagram.

Proof. Set $f = \overline{f}|_{B}$ and apply Remark 7.1.5.9 to the commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{ / f } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{ / (f \circ e) } \ar [d] \\ \operatorname{\mathcal{D}}_{ / (U \circ f) } \ar [r] & \operatorname{\mathcal{D}}_{ / (U \circ f \circ e) }, } \]

noting that the horizontal maps are equivalences by virtue of Proposition 7.2.2.1. $\square$