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For $F⊣G$, the counit is the natural transformation $\epsilon :FG\Rightarrow 1$ corresponding to the identity of $GY$ under the adjunction at each $Y$. It participates in the triangle identities for an adjunction.

If each standard counit presentation is a coequalizer, the Eilenberg-Moore comparison functor is full and faithful. An monad algebra morphism $\alpha :GB\to GC$ makes ${\epsilon }_{C}F\alpha$ equalize $FG{\epsilon }_B$ and ${\epsilon }_{FGB}$; its unique descent is the required morphism $B\to C$. The triangle identities for an adjunction make $G{\epsilon }_B$ a split epimorphism, so descent has underlying arrow exactly $\alpha$. Epimorphic cancellation at ${\epsilon }_B$ proves faithfulness.

The two formulas for the adjunction transposes give both triangle identities for an adjunction directly. Apply the inverse bijection to the transpose of $1_{FA}$:Apply the forward bijection to the inverse transpose of $1_{GB}$:These are equalities of components of natural transformations, soNo cancellation assumption on either functor is needed: both identities follow from the inverse hom-set bijections.

For $F⊣U$ with unit of an adjunction $\eta$ and counit of an adjunction $\epsilon$, putThe triangle identities for an adjunction giveFor associativity, naturality of $\epsilon$ at ${\epsilon }_{FA}:FUFA\to FA$ saysApply $U$ to obtain ${\mu }_{A}T({\mu }_A)={\mu }_A{\mu }_{TA}$. Thus this is the monad induced by an adjunction.

Conversely, start with a monad $(T,\eta ,\mu )$. The free algebra functor sends $A$ to $(TA,{\mu }_A)$ and $f$ to $Tf$. The monad laws ensure that these are algebras for a monad and morphisms of algebras for a monad. For an algebra $(B,b)$, there are inverse bijectionsThe inverse really is an algebra map becauseFor a free-algebra map $g$, its algebra equation and the monad unit law giveFor a base map $f$, naturality and the algebra unit law give $bT(f){\eta }_A=b{\eta }_{B}f=f$. The formulas are natural, establishing the free-forgetful Eilenberg-Moore adjunction $F^T⊣U^T$. Its unit is ${\eta }_A$, and its counit at $(B,b)$ is $b$. Hence its induced multiplication is $\mu$, and every monad arises from an adjunction.

Using the transpose formulas of the preceding part,Similarly,Thus the adjunction unit and adjunction counit satisfy both triangle identities for an adjunction,

Write the adjunction as a natural bijection ${\alpha }_{C,D}:\mathcal{D}(FC,D)\to \mathcal{C}(C,GD)$. The adjunction unit and adjunction counit are defined byNaturality of $\alpha$ implies $GF(u){\eta }_C={\eta }_{C^{\prime }}u$ for $u:C\to C^{\prime }$ and $v{\epsilon }_D={\epsilon }_{D^{\prime }}FG(v)$ for $v:D\to D^{\prime }$, so these are natural transformations $1_{\mathcal{C}}\to GF$ and $FG\to 1_{\mathcal{D}}$. More explicitly, naturality at $f:FC\to D$ and $g:C\to GD$ gives the two transpose formulasThese formulas will also prove the triangle identities for an adjunction.

Apply the inverse transpose formula to ${\eta }_A$, which by definition is the transpose of $1_{FA}$. This gives ${\epsilon }_{FA}F{\eta }_A=1_{FA}$. Likewise ${\epsilon }_B$ transposes to $1_{GB}$, so $G{\epsilon }_B{\eta }_{GB}=1_{GB}$. Hence the triangle identities for an adjunction areThese are equalities of natural transformations; their components are exactly the two identities just proved.

Write the adjunction unit as $\eta :1_{\mathcal{C}}\Rightarrow GF$ and the adjunction counit as $\epsilon :FG\Rightarrow 1_{\mathcal{D}}$. Define the monad induced by an adjunction bywith unit $\eta$. Naturality of $\epsilon$ makes $\mu$ a natural transformation.

The triangle identities for an adjunction give the two unit laws:For associativity, apply naturality of $\epsilon$ to the arrow ${\epsilon }_{FA}:FGFA\to FA$ in $\mathcal{D}$:Applying $G$ yieldsThus the induced data satisfy all the monad laws, rather than merely supplying an endofunctor and two transformations.

Write $\eta :1_{\mathcal{C}}\to GF$ and $\epsilon :FG\to 1_{\mathcal{D}}$ for the unit and counit of an adjunction. The induced monad hasThe Eilenberg-Moore comparison functor sends $D\in \mathcal{D}$ to the algebra for a monadand sends $v:D\to D^{\prime }$ to $Gv$. The triangle identities for an adjunction give the algebra unit law, and naturality of the counit gives the algebra multiplication law and the algebra-morphism equation for $Gv$.

If $\mathcal{D}$ has coequalizers, define the Left adjoint to the Eilenberg-Moore comparison functor at an algebra $(A,a:TA\to A)$ byThe parallel pair is reflexive through $F{\eta }_A$. A morphism $L(A,a)\to D$ is equivalently an arrow $b:FA\to D$ with $bFa=b{\epsilon }_{FA}$. Under $F⊣G$, its transpose is $\bar{b}=Gb{\eta }_A:A\to GD$. The transpose of $bFa$ is $\bar{b}a$, while the transpose of $b{\epsilon }_{FA}$ is $G{\epsilon }_{D}T\bar{b}$. Hence the coequalizer condition is exactlythe equation for a morphism of algebras for a monad $(A,a)\to K(D)$. This supplies the natural bijection proving $L⊣K$, and also defines $L$ on algebra morphisms by the coequalizer universal property.

A monad $\mathbb{T}=(T,\eta ,\mu )$ consists of an endofunctor $T:\mathcal{C}\to \mathcal{C}$ and natural transformations $\eta :1_{\mathcal{C}}\Rightarrow T$ and $\mu :T^2\Rightarrow T$ satisfyingAn algebra for a monad is $(A,a)$ with $a:TA\to A$ satisfying $a{\eta }_A=1_A$ and $a{\mu }_A=aT(a)$. A morphism of algebras for a monad $h:(A,a)\to (B,b)$ obeys $ha=bT(h)$. The functor laws show that identities and composites obey this equation, giving the Eilenberg-Moore category ${\mathcal{C}}^{\mathbb{T}}$.

For an adjunction $F⊣G$, let $\eta$ be its adjunction unit and $\epsilon$ its adjunction counit. The monad induced by an adjunction isThe two unit laws are $G{\epsilon }_{FA}{\eta }_{GFA}=1_{GFA}$ and $G({\epsilon }_{FA}F{\eta }_A)=1_{GFA}$, respectively the two triangle identities for an adjunction.

For associativity, naturality of the adjunction counit at ${\epsilon }_{FA}:FGFA\to FA$ saysApply $G$ to obtain ${\mu }_A{\mu }_{TA}={\mu }_{A}T{\mu }_A$. The composites defining $\eta$ and $\mu$ are natural transformations, so all monad axioms have been checked, not only their underlying objectwise types.

An adjunction $F⊣G$ is a family of bijectionsnatural in both variables. Its adjunction unit and adjunction counit are ${\eta }_A={\Phi }_{A,FA}(1_{FA})$ and ${\epsilon }_B={\Phi }_{GB,B}^{-1}(1_{GB})$. Naturality of the correspondence givesFor $f:A\to A^{\prime }$, naturality in both variables evaluates $\Phi (Ff)$ in two ways, giving $GFf{\eta }_A={\eta }_{A^{\prime }}f$. Thus $\eta$ is a natural transformation; the dual calculation gives naturality of $\epsilon$.

Applying the inverse correspondence to ${\eta }_A$ and the correspondence to ${\epsilon }_B$ yields the triangle identities for an adjunction:They express that transposing an identity morphism and transposing back returns that identity.

Write $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$. The characterization concerns an adjunction with the specified unit and counit of an adjunction $\eta$ and $\epsilon$. Its triangle identities for an adjunction areAssume these identities. Define the hom-set mapsNaturality of $\eta$ and $\epsilon$ makes these maps natural in both objects. Naturality and the triangle identities for an adjunction giveThus they are inverse bijections and define $F⊣G$.

Conversely, from the natural hom-set bijections of an adjunction, define ${\eta }_A=\Phi (1_{FA})$ and ${\epsilon }_D={\Phi }^{-1}(1_{GD})$. Naturality gives the same formulas for $\Phi$ and $\Psi$ above. Applying $\Psi \Phi$ to $1_{FA}$ and $\Phi \Psi$ to $1_{GD}$ gives the two triangle identities for an adjunction. Therefore these identities are exactly the compatibility conditions on the specified unit and counit. If the printed equivalence were read as mere existence of some adjunction, independently of the supplied transformations, its only-if direction would be false: on the category of abelian groups, $F=G=1$ are adjoint, but choosing both transformations to be zero does not satisfy either triangle on a nonzero object.

Put $e_A={\epsilon }_{FA}F{\eta }_A$. This is a natural transformation $F\Rightarrow F$. Only the $G$-triangle $G{\epsilon }_D{\eta }_{GD}=1_{GD}$ is assumed. Naturality gives two useful absorption identities:For the second equality in the last line use naturality of $\epsilon$ at ${\epsilon }_D$, and then the assumed $G$-triangle. Naturality of $\epsilon$ at $F{\eta }_A$ and of $\eta$ at ${\eta }_A$ now givesHence $e$ is an idempotent in the functor category: this is the one-triangle adjunction idempotent.

For the splitting of an idempotent morphism, suppose this idempotent morphism splits as natural transformations $r:F\Rightarrow H$ and $s:H\Rightarrow F$, with $sr=e$ and $rs=1_H$. DefineThe first absorption identity givesFor the other triangle, naturality of $s$ at ${\eta }_A^{\prime }$ and of $\epsilon$ at $r_A$ givesThus the triangle identities for an adjunction prove $H⊣G$.

Conversely, suppose $H⊣G$, with unit $\psi :1_{\mathcal{C}}\Rightarrow GH$ and counit $\phi :HG\Rightarrow 1_{\mathcal{D}}$. The unit has target $GH$, as its type requires. DefineThese are natural transformations. Transposition under $H⊣G$ gives $G{s}_A{\psi }_A={\eta }_A$. Independently, naturality of $\eta$ at ${\psi }_A$ and the assumed $G$-triangle giveThe transpose of $r_A{s}_A$ is therefore $G{r}_{A}G{s}_A{\psi }_A={\psi }_A$, the transpose of $1_{HA}$. Injectivity of the hom-set bijection implies $r_A{s}_A=1_{HA}$. Naturality of $\epsilon$ at $s_A$ givesConsequently $G$ has a left adjoint if and only if $e$ splits. This is the criterion for splitting a one-triangle adjunction idempotent. The argument gives both the explicit splitting and the new unit and counit, without assuming the other triangle for $F$.

Use the orientation $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$. An adjunction $F⊣G$ is equivalently specified by the natural transformationscalled the unit and counit of an adjunction, satisfying the triangle identities for an adjunctionThe corresponding natural bijection is $\mathcal{D}(FC,D)\cong \mathcal{C}(C,GD)$, with $f\mapsto G(f){\eta }_C$ and inverse $g\mapsto {\epsilon }_{D}F(g)$. The two triangular equations are the required compatibility conditions. No proof of equivalence of the formulations is needed here.

Write $K:\mathcal{B}\to \mathcal{C}$, $J:\mathcal{C}\to \mathcal{B}$, with adjunction unit $\rho$ and adjunction counit $\epsilon$. The adjunction givesFor $f:C\to C^{\prime }$, precomposition with ${\epsilon }_C$ followed by this bijection givesby the triangle identities for an adjunction. Thus $J$ is fully faithful exactly when precomposition with ${\epsilon }_C$ is bijective for every $C,C^{\prime }$.

A morphism $e:P\to Q$ with this property is an isomorphism. Surjectivity for target $P$ supplies $u:Q\to P$ with $ue=1_P$. Since $(eu)e=e=1_{Q}e$, injectivity for target $Q$ gives $eu=1_Q$. Conversely, precomposition with an isomorphism is always bijective. Applied to every ${\epsilon }_C$, this proves the fully faithful right-adjoint criterion:When the adjunction counit is invertible, the inverse to $f\mapsto Jf$ is explicitly $h\mapsto {\epsilon }_{C^{\prime }}K(h){\epsilon }_C^{-1}$.

Assume the hom-set condition. Its surjectivity at $B^{\prime }=B$ gives $u:KB\to B$ with $u{\rho }_B=1_B$. We show that this splitting is two-sided.

Because $J$ is fully faithful, ${\epsilon }_{KB}$ is invertible. Both triangle identities for an adjunction give, after suppressing $J$,Applying $K$ to $u{\rho }_B=1_B$ yields $Ku{\rho }_{KB}=1_{KB}$. Naturality of the adjunction unit at $u$ gives ${\rho }_{B}u=Ku{\rho }_{KB}$, hence ${\rho }_{B}u=1_{KB}$. Consequently the hom-set condition forces the unit to be invertible:Only surjectivity at $B^{\prime }=B$ was needed for this direction; the given family of bijections certainly supplies it.

The one-triangle adjunction idempotent splits if and only if $G$ has a left adjoint. From a splitting $F\stackrel{r}{\to }H\stackrel{s}{\to }F$, define the new unit $Gr\eta$ and counit $\epsilon s_G$; the absorption identities imply both triangle identities for an adjunction. Conversely, for $H⊣G$ with unit $\psi :1\Rightarrow GH$ and counit $\phi :HG\Rightarrow 1$, the splitting maps are $r={\epsilon }_{H}F\psi$ and $s={\phi }_{F}H\eta$. Their composites are $rs=1_H$ and $sr=e$.

For $F⊣G$, the unit is the natural transformation $\eta :1\Rightarrow GF$ corresponding to the identity of $FX$ under the adjunction at each $X$. It participates in the triangle identities for an adjunction.