colimit of the identity functor as terminal object

Define

as cocone.

This commutes as is the terminal morphism

Suppose is an object and we have a natural transformation

Then for each we have a factorization

Thus we may define

as a natural transformation

uniqueness follows as for each

Let be the colimti.
We claim that for each the terminal morhpism is given by the coprojection .

We have for each and that

commutes

In particular we may consider

and get

Then it follows that

are two morphisms of cocones.

It follows from uniqueness, that

Suppose is a morphism.
Then we have to show that

Here we use the diagram

which shows htat

Thus is a singleton