nLab
indexed functor

Idea

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).

Definition

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

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

Related concepts

• indexed adjoint functor theorem

References

Section B1 of

• Peter Johnstone, Sketches of an Elephant