## Ends I

So far in our treatment of enriched categories we’ve been working over a monoidal category , and we latter added the assumption that is symmetric and closed. From here, we’ll also assume that the underlying category is complete — it has all small limits.

Now let’s consider a functor . An “end” for is a universal -natural transformation . Universality here means that if is another \mathcal{V}$ natural transformation then there is a unique arrow so that . As usual, it’s unique up to isomorphism. We denote the object by , and abuse the language a bit by calling this object the end. Then we call the -natural transformation the “counit” of the end.

Because is symmetric and closed we have an adjunction . Under this adjunction, the two natural transformations

become transformations

Then the naturality condition states that .

Therefore when is small we can define the end as an equalizer:

where one of the arrows on the right is built from and the other is built from . This limit exists by the completeness of . In fact, a very similar argument can push the result a little further, to cover categories which are not small themselves, but which are *equivalent* to small categories.

We can write the universal property in yet another way. First, note that the set of -natural is in bijection with the set .

Now, we can write , and it turns out that this is *also* an end. Indeed, is -natural if and only if . Then there exists a unique , which corresponds under the closure adjunction to a unique . Thus , and thus the bijection of sets above gets promoted to an isomorphism in

[...] We continue our discussion of ends by noting that the process of taking an end (if it exists) is functorial in a [...]

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