Remark 5.6.6.16 (038N): Functoriality

Remark 5.6.6.16 (Functoriality). Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. We will see later that the corepresentable functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ and the representable functor $h_{Y}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ of Notation 5.6.6.14 depend functorially on the objects $X$ and $Y$, respectively. More precisely, the construction

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(\bullet , \bullet ): \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

can be promoted to a functor of $\infty $-categories $H: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ with the following properties:

For each object $X \in \operatorname{\mathcal{C}}$, the restriction $H|_{ \{ X\} \times \operatorname{\mathcal{C}}}$ is corepresentable by $X$.
For each object $Y \in \operatorname{\mathcal{C}}$, the restriction $H|_{ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \{ Y\} }$ is representable by $Y$.

See Proposition 8.3.3.2.