$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 8.3.1.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. Suppose that, for every object $Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is essentially small (this condition is satisfied, for example, if $\operatorname{\mathcal{C}}$ is small). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor, and let $\eta $ be a vertex of the Kan complex $\mathscr {F}(X)$. The following conditions are equivalent:
- $(1)$
The vertex $\eta $ exhibits the functor $\mathscr {F}$ as corepresented by $X$, in the sense of Definition 5.6.6.1.
- $(2)$
For every functor $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, the comparison map of Notation 8.3.1.2 is a homotopy equivalence $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}, \mathscr {G} ) \rightarrow \mathscr {G}(X)$.
- $(3)$
For every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, the comparison map of Notation 8.3.1.2 induces a bijection $\pi _0(\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}, \mathscr {G} )) \rightarrow \pi _0(\mathscr {G}(X))$.
Proof.
The implication $(1) \Rightarrow (2)$ follows from Proposition 8.3.1.3, and the implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3)$ implies $(1)$. Our assumption that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is essentially small for each $Y \in \operatorname{\mathcal{C}}$ guarantees that there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ and a vertex $\eta ' \in \mathscr {F}'(X)$ which exhibits $\mathscr {F}'$ as corepresented by $X$ (see Theorem 5.6.6.13). Applying assumption $(3)$, we deduce that there exists a natural transformation $\alpha : \mathscr {F}' \rightarrow \mathscr {F}$ such that $\alpha _{X}( \eta ' )$ and $\eta $ lie in the same connected component of $\mathscr {F}'$. Since the pair $(\mathscr {F}', \eta ')$ also satisfies condition $(3)$, composition with $\alpha $ induces a bijection $\pi _0( \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}, \mathscr {G} ) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}', \mathscr {G} ) )$ for each object $\mathscr {G} \in \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$. It follows that $\alpha $ is an isomorphism. Applying Remark 5.6.6.4, we deduce that $\alpha _{X}(\eta ') \in \mathscr {F}(X)$ exhibits the functor $\mathscr {F}$ as corepresented by $X$. Since $\eta $ and $\alpha _{X}(\eta )$ belong to the same connected component of $\mathscr {F}(X)$, it follows that $\eta $ has the same property (Remark 5.6.6.3).
$\square$