Let $\mathcal{C}$ be a category.
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1.
Functoriality. The assignment $\mathcal{C}\mapsto \mathsf{PSh}(\mathcal{C})$ defines a functor
\[ \mathsf{PSh}\colon \mathsf{Cats}\to \mathsf{Cats} \]up to some set-theoretic considerations.1
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2.
Interaction With Slice Categories. Let $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$. We have an equivalence of categories
\[ \mathsf{PSh}(\mathcal{C}_{/X})\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{PSh}(\mathcal{C})_{/h_{X}}. \] -
3.
Interaction With Categories of Elements. Let $\mathcal{F}\in \operatorname {\mathrm{Obj}}(\mathsf{PSh}(\mathcal{C}))$. We have an equivalence of categories
\[ \mathsf{PSh}(\textstyle \int _{\mathcal{C}}\mathcal{F})\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{PSh}(\mathcal{C})_{/\mathcal{F}}. \] -
1For instance:
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The $\mathsf{Cats}$ in the source of $\mathsf{PSh}$ could be small categories, and then the $\mathsf{Cats}$ in the right would be locally small categories.
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The $\mathsf{Cats}$ in the source of $\mathsf{PSh}$ could be locally small categories, and then the $\mathsf{Cats}$ on the right would be large categories.
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12.1.6 Properties of Categories of Presheaves
Proof of Proposition 12.1.6.1.1.
Item 1: Functoriality
Omitted.
Item 2: Interaction With Slice Categories
Omitted.
Item 3: Interaction With Categories of Elements
Omitted.
