nLab
enriched natural transformation

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)