Editing Adjoint functors (section)

===Composition===

Adjunctions can be composed in a natural fashion. Specifically, if 〈''F'', ''G'', ''ε'', ''η''〉 is an adjunction between ''C'' and ''D'' and 〈''{{prime|F}}'', ''{{prime|G}}'', ''{{prime|ε}}'', ''{{prime|η}}''〉 is an adjunction between ''D'' and ''E'' then the functor
:<math>F \circ F' : E \rightarrow C</math>
is left adjoint to
:<math>G' \circ G : C \to E.</math>
More precisely, there is an adjunction between ''F F&prime;'' and ''G&prime; G'' with unit and counit given respectively by the compositions:
:<math>\begin{align}
&1_{\mathcal E} \xrightarrow{\eta'} G' F' \xrightarrow{G' \eta F'} G' G F F' \\
&F F' G' G \xrightarrow{F \varepsilon' G} F G \xrightarrow{\varepsilon} 1_{\mathcal C}.
\end{align}</math>
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 category|small categories]] and whose morphisms are adjunctions.