# nLab enriched natural transformation

### Context

#### Enriched category theory

Could not include enriched category theory - contents

# Contents

## Idea

An enriched natural transformation is the appropriate notion of homomorphism between enriched functors, the analog in enriched category theory of the ordinary notion of natural transformation in ordinary category theory.

## Definition

Let $C$ and $D$ be categories enriched in a monoidal category $V$, and let $F, G \colon C \to D$ be enriched functors. We abbreviate hom-objects $\hom_C(c, d)$ to $C(c, d)$.

An enriched natural transformation $\eta \colon F \to G$ is a family of morphisms of $V$

$\eta_c \;\colon\; I \longrightarrow D(F c, G c)$

(out of the tensor unit $I$ of $V$) 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

For more references see at enriched category.

Revised on May 3, 2016 12:10:45 by Urs Schreiber (131.220.184.222)