A pseudofunctor $𝐹:\mathrm{ℬ︀}\to \mathrm{𝒞︀}$ is a lax functor whose unit and composition comparisons

are invertible 2-cells. So $𝐹$ preserves identities and composition up to coherent isomorphism.

Being pseudo is a property of a lax functor: invertibility of a 2-cell is a proposition, since inverses are unique. The data of a pseudofunctor is exactly the data of a lax functor, and everything proved about lax functors applies to pseudofunctors unchanged. Pseudofunctors are closed under composition and identities, as lax functors are, because invertible 2-cells are closed under composition and under the action of a functor on hom-categories.

The main examples here are prestacks, pseudofunctors ${\mathrm{ℬ︀}}^{\text{op}}\to \text{𝖢𝖠𝖳}$. When $\mathrm{ℬ︀}$ is locally discrete on a category $\mathrm{𝒞︀}$, these are the pseudofunctors ${\mathrm{𝒞︀}}^{\text{op}}\to \text{𝖢𝖠𝖳}$ of fibred category theory.