Remark 8.3.5.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. The datum of a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{\operatorname{Tw}(\operatorname{\mathcal{C}})}$ can be identified with a commutative diagram of $\infty $-categories
8.56
\begin{equation} \begin{gathered}\label{diagram:Hom-witness} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r] \ar [d] & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}\ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {H} } & \operatorname{\mathcal{S}}. } \end{gathered} \end{equation}
In this case, the natural transformation $\alpha $ exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ if and only if the diagram (8.56) is a categorical pullback square (see Corollary 5.1.7.15).