The category of presheaves $\mathsf{PSh}\webleft (\mathcal{C}\webright )$ and the category of copresheaves $\mathsf{CoPSh}\webleft (\mathcal{C}\webright )$ on a category $\mathcal{C}$ are the 1-categorical counterparts to the powerset $\mathcal{P}\webleft (X\webright )$ of subsets of a set $X$. The further analogies built upon this are summarised in the following table:
| Set Theory | Category Theory |
|---|---|
| Powerset $\mathcal{P}\webleft (X\webright )$ (Chapter 2: Constructions With Sets, Definition 2.4.1.1.1) | Presheaf category $\mathsf{PSh}\webleft (\mathcal{C}\webright )$ (  , ) |
| Characteristic function $\chi _{x}\colon X\to \{ \mathsf{t},\mathsf{f}\} $ (Chapter 2: Constructions With Sets, ) | Representable Presheaf $h_{X}\colon \mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}$ ( , of ) |
| Characteristic embedding $\chi _{\webleft (-\webright )}\colon X\hookrightarrow \mathcal{P}\webleft (X\webright )$ (Chapter 2: Constructions With Sets, of ) | Yoneda embedding ${\text{よ}}\colon \mathcal{C}\hookrightarrow \mathsf{PSh}\webleft (\mathcal{C}\webright )$ (, ) |
| Characteristic relation $\chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} $ (Chapter 2: Constructions With Sets, of ) | $\textup{Hom}$ profunctor $\textup{Hom}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )\colon \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\to \mathsf{Sets}$ ( ) |
| The Yoneda lemma for sets $\textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{U}\webright )\cong \chi _{U}\webleft (x\webright )$ (Chapter 2: Constructions With Sets, Proposition 2.5.5.1.1) | The Yoneda lemma for categories $\text{Nat}\webleft (h_{X},\mathcal{F}\webright )\cong \mathcal{F}\webleft (X\webright )$ (, ) |
| The characteristic embedding is fully faithful, $\textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{y}\webright )\cong \chi _{X}\webleft (x,y\webright )$ (Chapter 2: Constructions With Sets, Corollary 2.5.5.1.2) | The Yoneda embedding is fully faithful, $\text{Nat}\webleft (h_{X},h_{Y}\webright )\cong \textup{Hom}_{\mathcal{C}}\webleft (X,Y\webright )$ (, of ) |
| Subsets are unions of their elements, $\displaystyle U=\bigcup _{x\in U}\webleft\{ x\webright\} $ | Presheaves are colimits of representables, $\displaystyle \mathcal{F}\cong \operatorname*{\text{colim}}_{h_{X}\in \int _{\mathcal{C}}\mathcal{F}}\webleft (h_{X}\webright )$ () |
