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 $$F(f \circ g) = F(f) \circ F(g)$$ and $$F(1) = 1$$ do not hold as exact equalities but only up to coherent isomorphisms.

The Grothendieck construction associates to a pseudofunctor a fibered category.