# Contents

## Idea

An enriched natural transformation is the appropriate notion of morphism between functors enriched in a monoidal category $V$.

## Definition

Let $C$ and $D$ be categories enriched in a monoidal category $V$, and let $F, G: C \to D$ be enriched functors. We abbreviate hom-objects $\hom_C(c, d)$ to $C(c, d)$. An enriched natural transformation $\eta: F \to G$ is a family of morphisms of $V$

$\eta c: I \to D(F c, G c)$

indexed over $Ob(C)$, such that for any two objects $c$, $d$ of $C$ the following diagram commutes:

$\array{ C(c, d) & \stackrel{\rho}{\cong} & C(c, d) \otimes I & \stackrel{G_{c, d} \otimes \eta c}{\to} & D(G c, G d) \otimes D(F c, G c) \\ \stackrel{\lambda}{\cong} \downarrow & & & & \downarrow \circ_D \\ I \otimes C(c, d) & \underset{\eta d \otimes F_{c, d}}{\to} & D(F d, G d) \otimes D(F c, F d) & \underset{\circ_D}{\to} & D(F c, G d) }$

(Should expand to include other notions of enriched category.)

## Reference

• Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982) (pdf)

Revised on October 31, 2012 01:55:22 by Toby Bartels (64.89.53.173)