Corollary 11.5.0.46 (02W6)

Corollary 11.5.0.46. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, and suppose that the object $X$ belongs to a full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$: The morphism $f$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$ (see Definition 6.2.2.1).
$(2)$: The morphism $f$ is final when regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$.

In particular, an object $X \in \operatorname{\mathcal{C}}'$ is a $\operatorname{\mathcal{C}}'$-reflection of $Y \in \operatorname{\mathcal{C}}$ if and only if it represents the right fibration $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}'$.

Proof. Let us regard the object $Y \in \operatorname{\mathcal{C}}$ as fixed, and let $\theta : \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}'$ be the right fibration given by projection onto the first factor. Using Example 5.2.8.13, we can identify the enriched homotopy transport representation of $\theta $ with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $(X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}'}) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. The desired result now follows from the criterion of Proposition 5.6.6.21. $\square$