Pseudo-functor

In mathematics, a pseudofunctor F is a mapping between 2-categories, or from a category to a 2-category, that is just like a functor except that ${\displaystyle F(f\circ g)=F(f)\circ f(g)}$ and ${\displaystyle f(1)=1}$ do not hold as exact equalities but only up to coherent isomorphisms.

The Grothendieck construction associates to a pseudofunctor a fibered category.