A quick note: where is the terminal object.
Consider the following diagram,
we have already, and we attempt to find whether is equal to . However, where
Since is a terminal object, the morphism is unique. So, and therefore which is the identity of .
To see that is the identity, we just draw the following diagram,
where also makes the diagram commute. Since is unique, it must be as same as .
Also, to see we just draw the following diagram,
where by the similar argument.