Enriched Naturality Revisited

Let’s look back at the enriched versions of representable functors. If we fix an object we have a -natural transformation . This corresponds under the closure adjunction to . There’s a similar transformation for composition on the other side.

Now remember that the closest thing we have to a “morphism” in an enriched category is an element of the underlying set of a hom-object. That is, we can talk about an arrow . We often abuse the language and say that this is a morphism from to , which in fact it’s a morphism in the underlying category.

Now, even though this isn’t really a morphism in our enriched category, we can still come up with a morphism sensibly called . Here’s how it goes:

• We start with and use the left unit isomotphism to move to .
• We now hit with our morphism to land in .
• Finally, we compose to end up in .

We can similarly take and construct a morphism .

These composites should look familiar from the definition of enriched naturality for a transformation . In fact, we have a more compact diagram to replace that big hexagon:

Notice here that the right and bottom arrows in this square expand out to become the top and bottom of a hexagon, and we slip the functors and into place.

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