We summarise the analogies between un/straightening in set theory and category theory in the following table:
| Set Theory | Category Theory |
|---|---|
| Assignment $U\mapsto \chi _{U}$ | Assignment $\mathcal{F}\mapsto \int _{\mathcal{C}}\mathcal{F}$ (the category of elements) |
| Un/straightening isomorphism $\mathcal{P}\webleft (X\webright )\cong \mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright )$(Chapter 2: Constructions With Sets,  of ) | Un/straightening equivalence $\mathsf{PSh}\webleft (\mathcal{C}\webright )\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{DFib}\webleft (\mathcal{C}\webright )$( ) |
