Definition 7.5.7.3. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets restriction $\mathscr {F} = \overline{ \mathscr {F} }|_{\operatorname{\mathcal{C}}}$. We will say that $\overline{ \mathscr {F} }$ is a homotopy colimit diagram if the composite map
\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} ) \rightarrow \overline{\mathscr {F}}( {\bf 1} ) \]
is a weak homotopy equivalence of simplicial sets. Here ${\bf 1}$ denotes the final object of the cone $\operatorname{\mathcal{C}}^{\triangleright } \simeq \operatorname{\mathcal{C}}\star \{ {\bf 1}\} $, and the morphism on the left is the comparison map of Remark 5.3.2.9.