Adjunction

Let and be Categories
Let and be Functors
We say that is left adjoint to and is right adjoint to if

naturally in and . We write
An adjunction between and is this Isomorphism.
There are two other characterisations of adjunction:
Comma Category
Triangular Identities

what does naturally mean?

If and were Locally Small, we could express it as a Natural Isomorphism

as functors .
If they are not locally small we can express this in elementary terms as follows.
For in denote by the corresponding Morphism in
Similarly, for any denote by …

Naturality means that for any and we have

or equivalently

Note that I actually couldn’t write anything else sensible with these symbols.
This is because its the only “natural” thing to write down.

Corollary

If and are both left Adjoint to
then and are Isomorphic in

Proof

For any , and are both Initial objects
of the Comma Category
So there’s a unique Isomorphism .
Given , the composites and
are both morphisms in
so they’re equal.

Lemma

Suppose given and
with and .
Then

Proof

We have bijections
which are natural in both and .

Corollary

Suppose we are given a Commutative Diagram

in which all four Functors have left Adjoints.
Then the square of left adjoints commutes up to Natural Isomorphism.

Proof

The two ways round it are both left Adjoint to .
So they must be Isomorphic.