The basic analogies between set theory and category theory are summarised in the following table:
| Set Theory | Category Theory |
|---|---|
| Enrichment in $\{ \mathsf{true},\mathsf{false}\} $ | Enrichment in $\mathsf{Sets}$ |
| Set $X$ | Category $\mathcal{C}$ |
| Element $x\in X$ | Object $X\in \text{Obj}\webleft (\mathcal{C}\webright )$ |
| Function | Functor |
| Function $X\to \{ \mathsf{true},\mathsf{false}\} $ | Copresheaf $\mathcal{C}\to \mathsf{Sets}$ |
| Function $X\to \{ \mathsf{true},\mathsf{false}\} $ | Presheaf $\mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}$ |
