Kerodon

Notation 5.6.6.14 (Corepresentable Functors). Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. For every object $X \in \operatorname{\mathcal{C}}$, Theorem 5.6.6.13 asserts that there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresented by $X$, which is uniquely determined up to isomorphism. To emphasize this uniqueness, we will typically denote the functor $\mathscr {F}$ by $h^{X}$ and refer to it as the functor corepresented by $X$. For every object $Y \in \operatorname{\mathcal{C}}$, we can apply the same argument to the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to obtain a functor represented by $Y$, which we will typically denote by $h_{Y}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ and refer to as the functor represented by $Y$. Note that Remark 5.6.6.12 supplies isomorphisms $h^{X}(Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \simeq h_{Y}(X)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, depending functorially on the pair $(X,Y) \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.