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Lemma 8.3.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, let $\kappa $ be an uncountable cardinal, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor, and let $\eta \in \mathscr {F}(X)$ be a vertex. The following conditions are equivalent:
- $(1)$
The vertex $\eta $ exhibits $\mathscr {F}$ as corepresented by the object $X$, in the sense of Definition 5.6.6.1.
- $(2)$
Let $\iota : \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$ denote the inclusion map and let $\mathscr {F}_0: \{ X\} \rightarrow \operatorname{\mathcal{S}}$ denote the constant functor taking the value $\Delta ^0$, so that $\eta $ can be regarded as a natural transformation from $\mathscr {F}_0$ to the composite functor $\mathscr {F} \circ \iota $. Then $\eta $ exhibits $\mathscr {F}$ as a left Kan extension of $\mathscr {F}_0$ along $\iota $, in the sense of Variant 7.3.1.5. Moreover, for each object $Y \in \operatorname{\mathcal{C}}$, the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is essentially $\kappa $-small.
Proof.
Fix an object $Y \in \operatorname{\mathcal{C}}$ and set $M = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. We may assume without loss of generality that $M$ is essentially $\kappa $-small (this follows immediately from condition $(2)$, and also follows from $(1)$ since the Kan complex $\mathscr {F}(Y)$ is essentially small). For every Kan complex $K$, let $\underline{K}_{M}$ denote the constant functor $M \rightarrow \operatorname{\mathcal{S}}$ taking the value $K$, so that the functor $\mathscr {F}$ determines a natural transformation $\gamma : \underline{ \mathscr {F}(X) }_{M} \rightarrow \underline{ \mathscr {F}(Y) }_{M}$. We will show that the following pair of conditions is equivalent:
- $(1_ Y)$
The composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \mathscr {F}(X), \mathscr {F}(Y) ) \xrightarrow { \circ [\eta ] } \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {F}(Y) ) \]
is a homotopy equivalence of Kan complexes.
- $(2_ Y)$
The composite natural transformation
\[ \underline{ \Delta ^0 }_{M} \xrightarrow {\eta } \underline{ \mathscr {F}(X) }_{M} \xrightarrow {\gamma } \underline{ \mathscr {F}(Y) }_{M} \]
exhibits $\mathscr {F}(Y)$ as a colimit of the constant diagram $\underline{ \Delta ^0 }|_{M}$ in the $\infty $-category $\operatorname{\mathcal{S}}$.
The equivalence of $(1_ Y)$ and $(2_ Y)$ is a special case of Proposition 7.1.2.8 (see Example 7.1.2.10). Lemma 8.3.1.7 follows by allowing the object $Y \in \operatorname{\mathcal{C}}$ to vary.
$\square$