# Kerodon

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Remark 8.3.5.4. 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. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Notation 8.1.2.10 and Remark 5.6.1.5 supply canonical isomorphisms

$\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \quad \quad \mathscr {H}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {H}(X,Y) )$

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Consequently, the homotopy class of the morphism $\alpha _{X,Y}$ appearing in Definition 8.3.5.1 can be identified with a map $[ \alpha _{X,Y} ]: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \mathscr {H}(X,Y)$ in $\mathrm{h} \mathit{\operatorname{Kan}}$, which depends functorially on $X$ and $Y$ (see Corollary 8.1.2.14). 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) if and only if each $[ \alpha _{X,Y} ]$ is an isomorphism in the category $\mathrm{h} \mathit{\operatorname{Kan}}$.