Remark 11.5.0.11 (02XC)

Remark 11.5.0.11. See Corollary None.

Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets having restriction $f = \overline{f}|_{K}$. Proposition 7.1.6.13 asserts that $\overline{f}$ is a $U$-colimit diagram if and only if, for every diagram $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ having restriction $g = \overline{g}|_{K}$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( \overline{f}, \overline{g} ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f, g ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ \overline{f}, U \circ \overline{g} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f, U \circ g) } \]

is a homotopy pullback square. However, it suffices to verify this condition in the special case where $\overline{g}$ is a constant diagram: that is the content of Proposition 7.1.6.19.