# Kerodon

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Proposition 7.4.5.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ 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 corepresentable functor $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, the composite map $K \xrightarrow {f} \operatorname{\mathcal{C}}\xrightarrow {h} \operatorname{\mathcal{S}}$ has a contractible colimit.

Proof. 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 $f \circ h^{X}$ is a covariant transport representation for the left fibration $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} \rightarrow K$. Using Corollary 7.4.5.4, we can reformulate condition $(2)$ as follows:

$(2')$

For each object $X \in \operatorname{\mathcal{C}}$, the simplicial set $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ is weakly contractible.

The equivalence $(1) \Leftrightarrow (2')$ follows from Theorem 7.2.3.1. $\square$