An indexed category is a 2-presheaf. An indexed functor is a morphism of 2-presheaves.
The “indexed”-terminology here is traditional in 1-topos theory and hence indexed functors are usually considered only between pseudofunctors (as opposed to more general 2-functors).
Let be a category. Let and be -indexed categories, that is, pseudofunctors .
Then an -indexed functor is a pseudonatural transformation : it assigns to each object of a functor and to each morphism of a natural isomorphism that is coherent with respect to the structural isomorphisms of and (see pseudonatural transformation for details).
Section B1 of