Comma Category

Let , and be Categories
Let and
The comma category, written as
is the category defined as follows

1. Objects are triples with , and
2. maps are pairs
   such that we have a Commutative Diagram

Special case

Let be a Functor.
Let also be the functor from the trivial category to ,
sending the only element of to the object (by abuse of notation)
We write for their comma category.
This category has

1. Objects as pairs for and
2. Morphisms as morphisms
   such that we have a Commutative Diagram

Theorem

Let be a Functor.
Then specifying a left Adjoint for
is equivalent to
specifying an Initial object of the Comma Category
for each .

Proof

Suppose has a left Adjoint .
Let be the morphism corresponding to
Then is Initial in .
Let be an object in where
A map in is in
such that we have a Commutative Diagram

i.e.

But these correspond uniquely to

Thus exists and is unique.

Suppose for any we are given an initial object of
This defines a Functor on objects .
For any define to be the unique in
Then is a morphism
and functoriality of follows from uniqueness.
The Adjunction sends each
to the unique in .
On the other hand, each
is sent to and we can verify naturality.