A presheaf on $\mathcal{C}$ is a functor $\mathcal{F}\colon \smash {\mathcal{C}^{\mathsf{op}}}\to \mathsf{Sets}$.
12.1.1 Foundations
Let $\mathcal{C}$ be a category.
Presheaves on the delooping $\mathsf{B}{A}$ of a monoid $A$ are precisely the left $A$-sets; see 
, .
A morphism of presheaves on $\mathcal{C}$ from $\mathcal{F}$ to $\mathcal{G}$ is a natural transformation $\alpha \colon \mathcal{F}\Rightarrow \mathcal{G}$.
The category of presheaves on $\mathcal{C}$ is the category $\mathsf{PSh}(\mathcal{C})$1 defined by
\[ \mathsf{PSh}(\mathcal{C}) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{Fun}\big (\mathcal{C}^{\mathsf{op}},\mathsf{Sets}\big ). \]
- 1Further Notation: Also written $\widehat{\mathcal{C}}$ in some parts of the literature.
In detail, the category of presheaves on $\mathcal{C}$ is the category $\mathsf{PSh}(\mathcal{C})$ where
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Objects. The objects of $\mathsf{PSh}(\mathcal{C})$ are presheaves on $\mathcal{C}$ as in Definition 12.1.1.1.1.
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Morphisms. The morphisms of $\mathsf{PSh}(\mathcal{C})$ are morphisms of presheaves as in Definition 12.1.1.1.3, i.e. we have
\[ \operatorname {\mathrm{Hom}}_{\mathsf{PSh}(\mathcal{C})}(\mathcal{F},\mathcal{G})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Nat}}(\mathcal{F},\mathcal{G}) \]for each $\mathcal{F},\mathcal{G}\in \operatorname {\mathrm{Obj}}(\mathsf{PSh}(\mathcal{C}))$.
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Identities. For each $\mathcal{F}\in \operatorname {\mathrm{Obj}}(\mathsf{PSh}(\mathcal{C}))$, the unit map
\[ \mathbb {1}^{\mathsf{PSh}(\mathcal{C})}_{\mathcal{F}}\colon \mathrm{pt}\to \operatorname {\mathrm{Nat}}(\mathcal{F},\mathcal{F}) \]of $\mathsf{PSh}(\mathcal{C})$ at $\mathcal{F}$ is defined by
\[ \operatorname {\mathrm{id}}^{\mathsf{PSh}(\mathcal{C})}_{\mathcal{F}} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{\mathcal{F}}, \]where $\operatorname {\mathrm{id}}_{\mathcal{F}}\colon \mathcal{F}\Rightarrow \mathcal{F}$ is the identity natural transformation of Chapter 11: Categories, Example 11.9.3.1.1.
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Composition. For each $\mathcal{F},\mathcal{G},\mathcal{H}\in \operatorname {\mathrm{Obj}}(\mathsf{PSh}(\mathcal{C}))$, the composition map
\[ \circ ^{\mathsf{PSh}(\mathcal{C})}_{\mathcal{F},\mathcal{G},\mathcal{H}} \colon \operatorname {\mathrm{Nat}}(\mathcal{G},\mathcal{H}) \times \operatorname {\mathrm{Nat}}(\mathcal{F},\mathcal{G}) \to \operatorname {\mathrm{Nat}}(\mathcal{F},\mathcal{H}) \]of $\mathsf{PSh}(\mathcal{C})$ at $(\mathcal{F},\mathcal{G},\mathcal{H})$ is defined by
\[ \beta \circ ^{\mathsf{PSh}(\mathcal{C})}_{\mathcal{F},\mathcal{G},\mathcal{H}}\alpha \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\beta \circ \alpha , \]where $\beta \circ \alpha \colon \mathcal{F}\Rightarrow \mathcal{H}$ is the vertical composition of $\alpha $ and $\beta $ of Chapter 11: Categories, Definition 11.9.4.1.1.
