Kerodon

Example 7.4.3.7 (Colimits of Corepresentable Functors). Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor which is corepresentable by an object $X \in \operatorname{\mathcal{C}}$. Then the colimit $\varinjlim (\mathscr {F} )$ is contractible. To prove this, we observe that $\mathscr {F}$ is a covariant transport representation for the left fibration $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ (Proposition 5.6.6.21). By virtue of Proposition 7.4.3.6, it will suffice to show that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X/}$ is weakly contractible, which follows from Corollary 4.6.7.25.