structural:expansion, linguistic:paraphrasing
Adjunctions in full

There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.

An adjunction between categories C and D consists of

An equivalent formulation, where X denotes any object of C and Y denotes any object of D, is as follows:

From this assertion, one can recover that:

In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, the adjoint functors F and G alone are in general not sufficient to determine the adjunction. The equivalence of these situations is demonstrated below.