## Ends II

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

More specifically, let’s say we have two functors and from to , a -natural transformation , and that both of and have ends. Then we can compose with the transformation to get a -natural transformation from the end of to . By the universal property of there is then a unique arrow so that .

Now let’s consider a functor so that the end exists for each object . That is, we have arrows natural in for each . There is now a unique way of making into a functor so that is also natural in .

First, we note that is an end for . Indeed, we have

On the other hand, we have

and

By the universal property of the end , there is a unique arrow so that . These — along with the behavior of on objects — form a functor because they are -natural, and we can see this naturality from the litany of naturalities.

Now let’s consider a functor . If each end exists, then it they are the values of a -functor .

Further, every family that is -natural in factorizes uniquely into the composite of and . Again by our litany of naturalities, is -natural in if and only if is. Then since is natural in and separately if and only if it is natural in the pair , we deduce the “Fubini Theorem”: if every exists then

and either side exists if and only if the other side does.

Then, since we have the “interchange of ends” theorem: if every and every exists then

and either side exists if and only if the other side does.

It’s this formal, semantic similarity between the process of taking an end and the process of integration (which some of you may have heard of) that leads us to write an end as if it were an integral. We have bound variables ranging over categories, occurring covariantly and contravariantly, pairing off, and different variables do this essentially independently of each other.

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