Isomorphism

An isomorphism in a Category is a Morphism

such that there is an Inverse .
i.e. both of the following hold:

Lemma

Suppose is an isomorphism with inverse .
The following hold for any

Proof

Suppose
Compose with on the left:

Use Associativity

By definition of , we have

By definition of identities:

Now suppose
Note that in the Opposite Category

(where is composition in )
Thus, by previous case, we get