Proposition 7.4.3.11 (04K3)

Proposition 7.4.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$: The morphism $F$ is right cofinal (Definition 7.2.1.1).
$(2)$: For every functor of $\infty $-categories $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which admits a colimit $X = \varinjlim (G)$, the object $X \in \operatorname{\mathcal{D}}$ is also a colimit of the diagram $(G \circ F): \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}$.
$(3)$: For every uncountable regular cardinal $\kappa $ and every diagram $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ which admits a colimit $X = \varinjlim (G)$, the object $X \in \operatorname{\mathcal{S}}^{< \kappa }$ is also a colimit of the diagram $(G \circ F): \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$.
$(4)$: For every uncountable regular cardinal $\kappa $ and every corepresentable functor $G: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, the colimit $\varinjlim ( G \circ F)$ is contractible.

Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 7.2.2.11, the implication $(2) \Rightarrow (3)$ is immediate, and the implication $(3) \Rightarrow (4)$ follows from Example 7.4.3.7. We will complete the proof by showing that $(4)$ implies $(1)$. Choose an uncountable regular cardinal $\kappa $ such that $K$ is $\kappa $-small and $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Fix an object $X \in \operatorname{\mathcal{C}}$ and let $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor corepresented by $X$ (Theorem 5.6.6.13). Using Proposition 5.6.6.21, we see that $h^{X} \circ F$ is a covariant transport representation for the left fibration $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} \rightarrow K$. If condition $(4)$ is satisfied, then Proposition 7.4.3.6 guarantees that the simplicial set $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ is weakly contractible. Allowing $X$ to vary and applying the criterion of Theorem 7.2.3.1, we conclude that $F$ is right cofinal. $\square$