A copresheaf on $\mathcal{C}$ is a functor $F\colon \mathcal{C}\to \mathsf{Sets}$.
12.2.1 Foundations
Let $\mathcal{C}$ be a category.
Copresheaves on the delooping $\mathsf{B}{A}$ of a monoid $A$ are precisely the right $A$-sets; see 
, .
A morphism of copresheaves on $\mathcal{C}$ from $F$ to $G$ is a natural transformation $\alpha \colon F\Rightarrow G$.
The category of copresheaves on $\mathcal{C}$ is the category $\mathsf{CoPSh}(\mathcal{C})$ defined by
\[ \mathsf{CoPSh}(\mathcal{C}) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{Fun}(\mathcal{C},\mathsf{Sets}). \]
In detail, the category of copresheaves on $\mathcal{C}$ is the category $\mathsf{CoPSh}(\mathcal{C})$ where
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Objects. The objects of $\mathsf{CoPSh}(\mathcal{C})$ are copresheaves on $\mathcal{C}$ as in Definition 12.2.1.1.1.
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Morphisms. The morphisms of $\mathsf{CoPSh}(\mathcal{C})$ are morphisms of copresheaves as in Definition 12.2.1.1.3, i.e. we have
\[ \operatorname {\mathrm{Hom}}_{\mathsf{CoPSh}(\mathcal{C})}(F,G)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Nat}}(F,G) \]for each $F,G\in \operatorname {\mathrm{Obj}}(\mathsf{CoPSh}(\mathcal{C}))$.
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Identities. For each $F\in \operatorname {\mathrm{Obj}}(\mathsf{CoPSh}(\mathcal{C}))$, the unit map
\[ \mathbb {1}^{\mathsf{CoPSh}(\mathcal{C})}_{F}\colon \mathrm{pt}\to \operatorname {\mathrm{Nat}}(F,F) \]of $\mathsf{CoPSh}(\mathcal{C})$ at $F$ is defined by
\[ \operatorname {\mathrm{id}}^{\mathsf{CoPSh}(\mathcal{C})}_{F} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{F}, \]where $\operatorname {\mathrm{id}}_{F}\colon F\Rightarrow F$ is the identity natural transformation of Chapter 11: Categories, Example 11.9.3.1.1.
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Composition. For each $F,G,H\in \operatorname {\mathrm{Obj}}(\mathsf{CoPSh}(\mathcal{C}))$, the composition map
\[ \circ ^{\mathsf{CoPSh}(\mathcal{C})}_{F,G,H} \colon \operatorname {\mathrm{Nat}}(G,H) \times \operatorname {\mathrm{Nat}}(F,G) \to \operatorname {\mathrm{Nat}}(F,H) \]of $\mathsf{CoPSh}(\mathcal{C})$ at $(F,G,H)$ is defined by
\[ \beta \circ ^{\mathsf{CoPSh}(\mathcal{C})}_{F,G,H}\alpha \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\beta \circ \alpha , \]where $\beta \circ \alpha \colon F\Rightarrow H$ is the vertical composition of $\alpha $ and $\beta $ of Chapter 11: Categories, Definition 11.9.4.1.1.
