Today we’re going to talk about adjoint functors.
Definition: Let 
be a functor, and let 
be another functor. Then 
and 
are adjoint functors if for 
Obj
and 
Ob
, we have 
.
Maclane has famously stated in his seminal book that “adjoint functors arise everywhere”. However, what is the utility of such functors?
1. Solutions to Optimization Problems: Suppose you have an rng (which is a ring without the identity element), and you want to “adjoin” the minimal number of elements to the rng such that it becomes a ring. Explicitly, you want to adjoint 
to the rng, along with the elements 
for all 
. How can you solve this problem? For any fixed ring , consider the category 
in which the objects are morphisms of the form 
, where 
are unital rings. A morphism between the objects 
and is a morphism between 
and . Then the final object in this category is adjoined with the unity. This is the smallest ring which contains . Hence, this is the most efficient solution to our problem.
When we have a functor , we ask ourselves what is the problem to which is the most efficient solution. Then is the left adjoint of .
We have different formulations for adjoint functor. The unit-counit formulation says that 
and 
.
In terms of Hom isomorphism: An adjunction 
is a natural isomorphism between functors 
. What does this mean? We study the isomorphisms between functors from 
to 
. This natural isomorphism is given by 
. How is this a natural transformation between functors from to ? So we take a tuple 
, the make the following two functors act on it: 
and 
. Then the natural isomorphism is the isomorphism between these two functors. But why 
? Why not just 
? If one draws the commutative diagram, one will soon figure out that reversing the arrows in is a simple way of mapping 
to 
.
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