Remark 7.1.5.7 (02WZ)

Remark 7.1.5.7. Let $U_0: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $U_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which is isomorphic to $U_0$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then a diagram $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U_0$-limit diagram if and only if it is a $U_1$-limit diagram (see Remark 7.1.4.8). This follows by applying Remark 7.1.5.6 to each square of the diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}\ar [d]^{ U_0 } & \operatorname{\mathcal{C}}\ar [d]^{ U } \ar [l]_{\operatorname{id}} \ar [r]^-{\operatorname{id}} & \operatorname{\mathcal{C}}\ar [d]^{ U_1 } \\ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{D}}) & \operatorname{Isom}(\operatorname{\mathcal{D}}) \ar [l]_{ \operatorname{ev}_0 } \ar [r]^-{ \operatorname{ev}_{1} } & \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{D}}), } \]

where $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{Isom}(\operatorname{\mathcal{D}})$ classifies an isomorphism between $U_0$ and $U_1$; note that $\operatorname{ev}_0$ and $\operatorname{ev}_1$ are trivial Kan fibrations by virtue of Corollary 4.4.5.10.