---
tag: 18M8XGY04N
kind: concept
name: Pseudofunctor
msc: [18N10]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudo-functor
same_as:
  wikidata: Q16919007
  wikipedia: Pseudo-functor
---

## Summary

A pseudofunctor is a map between [[bicategories|TAHV24ED74]] that preserves composition and identities up to specified invertible 2-morphisms, rather than on the nose.
These comparison 2-morphisms must be compatible with the associators and unitors.
Pseudofunctors into the 2-category of categories describe families of categories that vary with a parameter.
Examples are the categories of modules over varying rings, or of sheaves over varying spaces.
The [[Grothendieck construction|272T16NWZ9]] turns such a pseudofunctor into a fibred category.

## Explanations

### L1 human

A functor must send a composite $g \circ f$ to exactly $F(g) \circ F(f)$.
In many natural constructions this holds only up to a canonical isomorphism.
For example, extending scalars along $R \to S$ and then along $S \to T$ gives modules isomorphic, but not equal, to extending along $R \to T$ in one step.
A pseudofunctor keeps these isomorphisms as data and asks them to fit together consistently.