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).

Definition

Let $S$ be a category. Let $\mathbb{C}$ and $\mathbb{D}$ be $S$-indexed categories, that is, pseudofunctors$S^\mathrm{op} \to Cat$.

Then an $S$-indexed functor$F:{\mathbb{C}}\to{\mathbb{D}}$ is a pseudonatural transformation$F \colon \mathbb{C} \Rightarrow \mathbb{D}$: it assigns to each object$A$ of $S$ a functor$F^A:{\mathbb{C}}^A\to{\mathbb{D}}^A$ and to each morphism$f:A\to B$ of $S$ a natural isomorphism$\mathbb{D}(f) \circ F^B \cong F^A \circ \mathbb{C}(f)$ that is coherent with respect to the structural isomorphisms of $\mathbb{C}$ and $\mathbb{D}$ (see pseudonatural transformation for details).