# Pseudo-functor

In mathematics, a pseudofunctor f is a mapping between categories that is just like a functor except that ${\displaystyle f(x\circ y)=f(x)\circ f(y)}$ 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.