$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 8.3.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor, and let $\alpha : \underline{ \Delta ^{0} }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ be a natural transformation. The following conditions are equivalent:
- $(1)$
The natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.3.5.1).
- $(2)$
The natural transformation $\alpha $ exhibits the profunctor $\mathscr {H}$ as represented by the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 8.3.4.12). That is, for every object $X \in \operatorname{\mathcal{C}}$, the vertex $\alpha ( \operatorname{id}_{X} ) \in \mathscr {H}(X,X)$ exhibits the functor $\mathscr {H}(-, X): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by the object $X$.
- $(3)$
The natural transformation $\alpha $ exhibits the profunctor $\mathscr {H}$ as corepresented by the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Variant 8.3.4.16). That is, for every object $X \in \operatorname{\mathcal{C}}$, the vertex $\alpha ( \operatorname{id}_{X} ) \in \mathscr {H}(X,X)$ exhibits the functor $\mathscr {H}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by the object $X$.
Proof.
We will show that $(1) \Leftrightarrow (3)$; the proof of the equivalence $(1) \Leftrightarrow (2)$ is similar. The natural transformation $\alpha $ can be identified with a functor $T: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}$. For each object $X \in \operatorname{\mathcal{C}}$, let $T_{X}$ denote the restriction of $B$ to the simplicial subset $\{ X \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Tw}(\operatorname{\mathcal{C}})$, and consider the following condition:
- $(1_ X)$
The diagram of $\infty $-categories
8.57
\begin{equation} \begin{gathered}\label{diagram:Hom-witness-later} \xymatrix@R =50pt@C=50pt{ \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T_ X} \ar [d] & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}\ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {H}(X, -) } & \operatorname{\mathcal{S}}} \end{gathered} \end{equation}
is a categorical pullback square.
By virtue of Corollary 5.1.7.15, the natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if and only if it satisfies condition $(1_ X)$ for every object $X \in \operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that $(1_ X)$ is satisfied if and only if $\alpha (\operatorname{id}_ X) \in \mathscr {H}(X,X)$ exhibits the functor $\mathscr {H}(X, -)$ as corepresented by $X$. This is a special case of Proposition 5.6.6.21, since the $\operatorname{id}_{X}$ is an initial object of the $\infty $-category $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ (Proposition 8.1.2.1).
$\square$