Kerodon

Example 7.4.3.5. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be the constant diagram $\underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}$ taking the value $\Delta ^0 \in \operatorname{\mathcal{S}}$. Then $\mathscr {F}$ is a covariant transport representation for the left fibration $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$. In this case, Proposition 7.4.3.1 asserts that a Kan complex $X$ is a colimit of $\mathscr {F}$ if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{C}}\rightarrow X$, which is a special case of Example 7.1.2.10.