# 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 ${\mathrm{hom}}_{C}(c,d)$ to $C(c,d)$. An **enriched natural transformation** $\eta :F\to G$ is a family of morphisms of $V$

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

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

$$\begin{array}{ccccc}C(c,d)& \stackrel{\rho}{\cong}& C(c,d)\otimes I& \stackrel{{G}_{c,d}\otimes \eta c}{\to}& D(Gc,Gd)\otimes D(Fc,Gc)\\ \stackrel{\lambda}{\cong}\downarrow & & & & \downarrow {\circ}_{D}\\ I\otimes C(c,d)& \underset{\eta d\otimes {F}_{c,d}}{\to}& D(Fd,Gd)\otimes D(Fc,Fd)& \underset{{\circ}_{D}}{\to}& D(Fc,Gd)\end{array}$$```
\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)