Pseudo-functor

In mathematics, a pseudofunctor F is a mapping from a category to the category Cat of categories that is just like a functor except that and do not hold as exact equalities but only up to coherent isomorphisms.
A typical example is an assignment to each pullback, which is a contravariant
pseudofunctor since, for example for a quasi-coherent sheaf, we only have:
Since Cat is a 2-category, more generally, one can also consider a pseudofunctor between 2-categories, where coherent isomorphisms are given as invertible 2-morphisms.
The Grothendieck construction associates to a contravariant pseudofunctor a fibered category, and conversely, each fibered category is induced by some contravariant pseudofunctor. Because of this, a contravariant pseudofunctor, which is a category-valued presheaf, is often also called a prestack.

Definition

A pseudofunctor F from a category C to Cat consists of the following data

a category for each object x in C,
a functor for each morphism f in C,
a set of coherent isomorphisms for the identities and the compositions; namely, the invertible natural transformations
:,
: for each object ''x''

The notion of a pseudofunctor is more efficiently handled in the language of higher category theory. Namely, given an ordinary category C, we have the functor category as the ∞-category
Each pseudofunctor belongs to the above, roughly because in an ∞-category, a composition is only required to hold weakly, and conversely.