Category Theory III 7.1, Natural transformations as ends

[@prontopuntor]

On 24:19 f is h, id is j. In dimap we always pre-compose first function (contravariant, "backward") and post-compose second (covariant, "forward"). I keep in mind an intuition: if we have old storage of values by keys, then new storage can be constructed as: 1) translate new key to old key ("backward function") 2) using old key get old value from old storage 3) translate old value to new value ("forward function")

[@clarejang9088]

Shouldn't the `C` in the end notation in 28:50 be `D`? `f` and `g` are functors from `C` to `D`, so I think the result objects `f a` and `g a` should be objects in `D`.

[@clarejang9088]

I had to watch more. He fixed it in 31:57

[@DrBartosz]

If you keep watching, I correct it later.

[@DanOneata]

Very nice presentation - as always! :-) I have one question: you showed that the natural transformation from f to g is a wedge of the functor D(f -, g =), but in order to show it corresponds to an end, wasn't there one more step required: show that the wedge is terminal?

[@DrBartosz]

Good question. I concentrated on the crux of the proof, but there is more to it. When you write the universal condition, you pick a set X and a wedge from it to D(f-,g=). For every element of X, this wedge picks a family of morphisms tau. I have shown that this tau is natural. Conversely, if you want to show that every natural transformation is covered this way, use the singleton set for X.