linguistic:translation
Composition

Adjunctions can be composed in a natural fashion. Specifically, if 〈 F, G, ε, η〉 is an adjunction between C and D and 〈 F ′, G ′, ε′, η′〉 is an adjunction between D and E then the functor

is left adjoint to

More precisely, there is an adjunction between F F' and G' G with unit and counit given respectively by the compositions:

This new adjunction is called the composition of the two given adjunctions.

Since there is also a natural way to define an identity adjunction between a category C and itself, one can then form a category whose objects are all small categories and whose morphisms are adjunctions.