Adjunction is a special relationship between two functors going in the opposite direction, and it occurs in many mathematical cases. It says that the one-way relation from $F(A)$ to $B$ is identical to the relation from $A$ to $G(B)$ naturally, which implies that the specific structure is interchangeable between the two categories. In this section:

1. Defines this concept in a concrete manner by an object’s scale,
2. explains an equivalent definition that is useful in many categorical proofs,
3. and gives another equivalent definition that shows the concept’s occurrence in many mathematical cases.

Adjunctions

A strict definition of an adjunction is the one-to-one correspondences $\mathcal{B}(F(A),\,B)\cong\mathcal{A}(A,\,G(B))$ for all $A$ and $B$, with a naturality axiom. This makes the naturality between adjunction and composition. If the adjunction exists, then $F$ is left adjoint to $G$, and $G$ is right adjoint to $F$. Also, a morphism corresponds to morphism $f$ by the adjunction is called the transpose of $f$, denoted by $\bar{f}$.

Below are some examples and applications of adjunction:

1. The most understandable example of the adjunction would be the case between the free functor and the forgetful functor. The free functor makes an expansion to the codomain category, considering each element in the original object as an independent individual, while the forgetful functor just removes some properties from the domain category. We study this concept with a universal property, and later we can recognize the universal property as the condition for the alternative definition of the adjunction. Also, much later, we prove that the forgetful functor and the functor like this must have a left adjoint(GAFT).
2. If the category is a subcategory of another category, there exists an inclusion functor that goes from the former to the latter. Using the concept of adjunction, we define some specific kinds of subcategories:
   • reflective - inclusion functor has a left adjoint,
   • coreflective - inclusion functor has a right adjoint.
3. Initial and terminal objects can be described as the language of adjunction. The initial object is the object that for any object in the category, there exists a unique morphism that goes from the initial object to that object, and the terminal object is defined dually. We know that the functor from arbitrary category to $\mathbb{1}$(consisting of 1 object and identity morphism only) exists uniquely, and its left/right adjoint corresponds to the initial/terminal object of the category, respectively(and is easy to check).

Adjunctions via Units and Counits

This subsection describes the alternative definition of the adjunction, with units and counits. Given the adjoint relation, the unit is the collection of $\overline{1_{F(A)}}$ for all $A\in \mathcal{A}$, and the counit is defined dually. They have certain properties derived by properties of adjunction:

• By the naturality of adjunction, they are actually natural transformations.
• Since the adjunction is a one-to-one correspondence, they satisfy some triangle identities below.

In addition, since the identities mentioned above could be looked sophisticated at first look because they are the commute diagrams in the functor category, this book gives another way to interpret this, which is called a string diagram. This can show the movement of categories by each functor as well as the movement of functors by each natural transformation, so it helps us not to be lost in the forest by the tree.

This unit and counit only show a few relations inside the adjunction. Actually, in converse, we can construct a complete adjunction from two corresponding natural transformations satisfying the properties. Thus, defining unit and counit is equivalent to defining adjunction, which is the main theorem of this subsection. The major intuition and implication of the theorem are that unit and counit allow us to add or remove the action of $FG$ and $GF$ on objects with great naturality, which is induced from the triangle identities. Therefore, we can operate the functor on morphism and then adjust corresponding objects by unit and counit to make a one-to-one correspondence.

Adjunctions via Initial Objects

This subsection shows another alternative definition of the adjunction, by only the unit with the better property, which arises as the universal property in other mathematical areas. Furthermore, this subsection gives another interpretation of this property using the comma category, which considers the morphism as an object and the commute diagram as a morphism.

More specifically, for the given functors $P:\mathcal{A}→\mathcal{C}$ and $Q:\mathcal{B}→\mathcal{C}$, the comma category $(P⇒Q)$(or $(P\downarrow Q)$) is defined as follows: