Triangular Identities

Given Functors and ,
specifying an Adjunction
is equivalent to specifying Natural Transformations
and satisfying

the triangular identities.

Proof

Given , let and be Unit and Counit respectively.
Then corresponds under the Adjunction to , so it’s .
The second identity is dual.
Conversely, suppose given and satisfying triangular identities.
Given we define to be
and given we define to be
Then is

so and dually
And and are natural since and are.

Proposition

Suppose given an equivalence and
and and Natural Isomorphisms
and
Then there are Natural Isomorphisms and
satisfying the Triangular Identities.
In particular and .

Proof

We define and take to be

Note that since

is a Commutative Diagram by naturality and is Monomorphism.
Similarly
The triangular identities for and are

and

so and are the unit and counit of Adjunction
and and are the unit and counit of .