There are a bunch of natural (in both senses) maps we can consider now. Some look all but tautological over , and we may have used them in the past without comment. In the enriched context, though, we should go over them.
- The family of arrows is extraordinarily -natural exactly when is -natural.
- For a -functor , the map is natural in both and .
- Similarly, if is a -functor, then is -natural. And it’s also natural in .
- In particular, is natural in all three variables.
- Putting together naturalities 1 and 4 tells us that for a morphism , the transformation is natural.
- The “evaluation” is natural in both variables.
- Since we built compositions from evaluations and the closure adjunction, the arrow is natural in all variables.
- The identity functor on has an identity natural transformation, so by naturality 1 we see that is natural.
- All the monoidal structural isomorphisms — , , , and — are natural in all variables.
- We can start with and hit it with to get . Then we can evaluate to get . This is an isomorphism we call , corresponding to the adjunction , and it’s natural in all variables.
- We can compose the following arrows:
and we get the “coevaluation” — the counit of the closure adjunction. And thus the coevaluation is -natural in all variables.
- We can compose the coevaluation , and then use the right unit isomorphism to get a -natural isomorphism .
- In general, a family is -natural in any of its variables if the corresponding variables are natural in
Whew. That’s a mouthful. It can be instructive to sit down and try to interpret some of these in the context of categories enriched over , so when you recover I’d advise taking a look at that.