Let $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$.
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The corepresentable copresheaf associated to $A$ is the copresheaf
\[ h^{A}\colon \mathcal{C}\to \mathsf{Sets} \]where
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Action on Objects. For each $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, we have
\[ h^{A}(X) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,X). \] -
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Action on Morphisms. For each $X,Y\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the action on morphisms
\[ h^{A}_{X,Y}\colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(X,Y)\to \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(h^{A}(X),h^{A}(Y)) \]of $h^{A}$ at $(X,Y)$ is given by sending a morphism
\[ f\colon X\to Y \]of $\mathcal{C}$ to the map of sets
\[ h^{A}(f) \colon \underbrace{h^{A}(X)}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,X)} \to \underbrace{h^{A}(Y)}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,Y)} \]defined by
\[ h^{A}(f) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{*}, \]where $f_{*}$ is the postcomposition by $f$ morphism of Chapter 11: Categories, Item 2 of Definition 11.1.4.1.1.
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2.
A corepresenting object for a copresheaf $F\colon \mathcal{C}\to \mathsf{Sets}$ on $\mathcal{C}$ is an object $A$ of $\mathcal{C}$ such that we have $F\cong h^{A}$.
