Wikipedia
Pseudofunctor
In mathematics, a pseudofunctor f is a mapping between categories that is just like a functor except that f(x ∘ y) = f(x) ∘ f(y) 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.