We summarise the analogies between functions $\mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright )$ and functors $\mathsf{PSh}\webleft (\mathcal{C}\webright )\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$ in the following table:
| Set Theory | Category Theory |
|---|---|
| Direct image function $f_{*}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright )$ (Chapter 2: Constructions With Sets, Definition 2.6.1.1.1) | Direct image functor $F_{*}\colon \mathsf{PSh}\webleft (\mathcal{C}\webright )\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$ ( ) |
| Inverse image function $f^{-1}\colon \mathcal{P}\webleft (Y\webright )\to \mathcal{P}\webleft (X\webright )$ (Chapter 2: Constructions With Sets, Definition 2.6.2.1.1) | Inverse image functor $F^{*}\colon \mathsf{PSh}\webleft (\mathcal{D}\webright )\to \mathsf{PSh}\webleft (\mathcal{C}\webright )$ () |
| Direct image with compact support function $f_{!}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright )$ (Chapter 2: Constructions With Sets, Definition 2.6.3.1.1) | Direct image with compact support functor $F_{!}\colon \mathsf{PSh}\webleft (\mathcal{C}\webright )\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$ () |
