Editing Adjoint functors (section)

===Definition via counit–unit===

A third way of defining an adjunction between two categories <math>\mathcal{C}</math> and <math>\mathcal{D}</math> consists of two [[functor]]s <math>F : \mathcal{D} \to \mathcal{C}</math> and <math>G : \mathcal{C} \to \mathcal{D}</math> and two [[natural transformation]]s
:<math>\begin{align}
\varepsilon &: FG \to 1_\mathcal{C} \\
\eta &: 1_\mathcal{D} \to GF\end{align}</math>
respectively called the '''counit''' and the '''unit''' of the adjunction (terminology from [[universal algebra]]), such that the compositions
:<math>F\xrightarrow\overset{}{\;F\eta\;}FGF\xrightarrow\overset{}{\;\varepsilon F\,}F</math>
:<math>G\xrightarrow\overset{}{\;\eta G\;}GFG\xrightarrow\overset{}{\;G \varepsilon\,}G</math>
are the identity morphisms <math>1_F</math> and <math>1_G</math> on {{mvar|F}} and {{mvar|G}} respectively.

In this situation we say that {{mvar|F}} '''is left adjoint to''' {{mvar|G}} and {{mvar|G}} '''is right adjoint to''' {{mvar|F}}, and may indicate this relationship by writing &nbsp;<math>(\varepsilon,\eta):F\dashv G</math>&nbsp;, or, simply &nbsp;<math>F\dashv G</math>&nbsp;.

In equational form, the above conditions on <math>(\varepsilon,\eta)</math> are the '''counit–unit equations'''
:<math>\begin{align}
1_F &= \varepsilon F\circ F\eta\\
1_G &= G\varepsilon \circ \eta G
\end{align}</math>
which imply that for each <math>X \in \mathcal{C}</math> and each <math>Y \in \mathcal{D},</math>
:<math>\begin{align}
1_{FY} &= \varepsilon_{FY}\circ F(\eta_Y) \\
1_{GX} &= G(\varepsilon_X)\circ\eta_{GX}
\end{align}</math>.

Note that <math>1_{\mathcal C}</math> denotes the identify functor on the category <math>\mathcal C</math>, <math>1_F</math> denotes the identity natural transformation from the functor {{mvar|F}} to itself, and <math>1_{FY}</math> denotes the identity morphism of the object <math>FY.</math>
[[File:String diagram adjunction.svg|thumb|String diagram for adjunction.]]
These equations are useful in reducing proofs about adjoint functors to algebraic manipulations.  They are sometimes called the ''triangle identities'', or sometimes the ''zig-zag equations'' because of the appearance of the corresponding [[string diagram]]s.  A way to remember them is to first write down the nonsensical equation <math>1=\varepsilon\circ\eta</math> and then fill in either {{mvar|F}} or {{mvar|G}} in one of the two simple ways that make the compositions defined.

Note: The use of the prefix "co" in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies an ''initial'' property whereas the counit morphisms  satisfy ''terminal'' properties, and dually for limit versus unit.  The term ''unit'' here is borrowed from the theory of [[Monad (category theory)|monads]], where it looks like the insertion of the identity {{math|1}} into a [[monoid]].