There is a tight relationship between monads and adjunctions. In fact, it is easy to see that every pair of adjoint functors yields a monad. What is really interesting is that conversely, every monad arises by some adjunction.

Let’s look at an example first. Fix a set \(S\), then there is a natural bijection of hom-sets in the category \(\cat{Set}\):

\[c: \hom(X\times S, Y)\xrightarrow{\cong}\hom(X, \hom(S, Y)),\]

known as “currying”. It sends a function \(f : X\times S\to Y\) to a “curried” function \(c(f) : X\to \hom(S, Y)\) defined by \(c(f)(x) = s\mapsto f(x,s)\). This can be interpreted as an adjunction, where \(-\times S\) is the left adjoint and \(\hom(S, -)\) is the right adjoint. The composition of these functors is the functor \(X\mapsto \hom(S, X\times S)\), which you might recognize as the State monad.

Adjoint functors

We recall the definition of an adjoint pair of functors and some properties (without proof.) Let \(C\), \(D\) be categories, \(F: C\to D\) and \(G: D\to C\) be functors. Then \(F\), \(G\) is called an adjoint pair if there is a natural bijection

\[\Phi_{XY} : \hom_D(FX, Y)\xrightarrow{\cong}\hom_C(X, GY).\]

We call \(F\) the left-adjoint and \(G\) the right-adjoint, and we write \(F\dashv G\).

An adjunction comes with natural transformations, \(\eta: G\circ F\natto\Id\), the unit of the adjunction, and \(\varepsilon: \Id\natto F\circ G\), the counit. These arise as follows:

\[\eta_X = \Phi_{X,FX}(\id_{FX}) : X\to GFX,\]

\[\varepsilon_Y = \Phi_{GY,Y}^{-1}(\id_{GY}) : FGY\to Y,\]

and they satisy so-called triangle equations given by the following commuting diagrams:

\[\require{AMScd} \begin{CD} F\circ\Id_C @>{F\bullet\eta}>> F\circ G\circ F\\ @| @V{\varepsilon\bullet F}VV\\ F @= \Id_D\circ F \end{CD}\]

\[\begin{CD} \Id_C\circ G @>{\eta\bullet G}>> G\circ F\circ G\\ @| @V{G\bullet \varepsilon}VV\\ G @= G\circ \Id_D \end{CD}\]

(Forgive me if these don’t show up as actual triangles due to technical limitations.)

It will come in handy later to write \(\Phi\) in terms of \(\eta\) and \(\varepsilon\):

\[\begin{align*} \Phi_{XY}(f : FX\to Y) &= G(f)\circ\eta_X &: X \xrightarrow{\eta_X} GFX \xrightarrow{Gf} GY\\ \Phi^{-1}_{XY}(g :X\to GY) &= \varepsilon_Y\circ F(g) &: FX \xrightarrow{Fg} FGY \xrightarrow{\varepsilon_Y} Y \end{align*}\]

Monads from adjunctions

When we post-compose the first triangle diagram with \(G\) and pre-compose the second with \(F\), we obtain a very familiar diagram:

\[\begin{CD} \Id_C\circ GF @>{\eta\bullet GF}>> GFGF @<{GF\bullet\eta}<< GF\circ\Id_C\\ @| @V{G\bullet\varepsilon\bullet F}VV @|\\ GF @= GF @= GF \end{CD}\]

Set \(M := G\circ F\) and \(\mu := G\bullet\varepsilon\bullet F\). We obtain the following diagram:

\[\begin{CD} \Id_C\circ M @>{\eta\bullet M}>> M\circ M @<{M\bullet \eta}<< M\circ\Id_C\\ @| @V{\mu}VV @| \\ M @= M @= M \end{CD}\]

This ist just the diagram for the left and right identities for monads! To see that associativity holds, first recall the horizontal composition \(\varepsilon\bullet\varepsilon : FGFG\natto\Id_D\). It was defined as the diagonal of this commuting diagram:

\[\begin{CD} FGFG @>{FG\bullet\varepsilon}>> FG\circ\Id_D\\ @V{\varepsilon\bullet FG}VV @V{\varepsilon\bullet\Id_D}VV\\ \Id_D\circ FG @>{\Id_D\bullet\varepsilon}>> \Id_D\circ\Id_D \end{CD}\]

Post-compose with \(G\) and pre-compose with \(F\), similar to what we did before:

\[\begin{CD} GFGFGF @>{GFG\bullet\varepsilon\bullet F}>> GFGF\\ @V{G\bullet\varepsilon\bullet FGF}VV @V{G\bullet\varepsilon \bullet F}VV\\ GFGF @>{G\bullet\varepsilon\bullet F}>> GF \end{CD} \quad \sim \quad \begin{CD}MMM @>{M\bullet\mu}>> MM\\ @V{\mu\bullet M}VV @V{\mu}VV\\ MM @>{\mu}>> M \end{CD}\]

This is exactly the associativity diagram we need.

Conclusion so far: Every adjunction yields a monad.

There’s a dual statement included that I won’t go into in this article. Every adjunction also yields a comonad \(W = F\circ G\), with the given \(\varepsilon: \Id\natto W\) and \(\Delta = F\bullet\eta\bullet G : W\natto W\circ W\).

Let’s see about the converse statement. Does every monad come from an adjunction?

Adjunctions from monads

There are two very different canonical ways to find an adjoint pair that form a monad. In the following, let \((M,\eta,\mu)\) be a monad in a category \(C\).

The Kleisli category

Let \(C_M\) be the category whose objects are exactly the objects of \(C\), and whose arrows from \(X\) to \(Y\) are exactly the \(C\)-morphisms \(X\to MY\). To avoid confusion, we’re going to write \(X\to_M Y\) for the arrows in \(C_M\) and \(X\to Y\) for arrows in \(C\).

Given arrows \(f: X\to MY\), \(g: Y\to MZ\), an arrow \(g\circ_M f: X\to MZ\) is given by the Kleisli composition, \(g\circ_M f := \mu_Z\circ M(g)\circ f\).

\[ g\circ_M f := X\stackrel{f}\longrightarrow MY\stackrel{M(g)}{\longrightarrow} MMZ\stackrel{\mu_Z}{\to} MZ\]

We check that this results in a lawful composition.

1. \(\eta_Y\circ_M f = \mu_Y\circ M(\eta_Y)\circ f = f\), because \(\mu\circ(M\bullet\eta) = \id_M\) (this is, perhaps confusingly, the right identity law for monads.)

2. \(f\circ_M\eta_X = \mu_Y\circ M(f)\circ\eta_X = \mu_Y\circ\eta_{MY}\circ f = f\) by naturality and because \(\mu\circ(\eta\bullet M) =\id_M\) (this is the left identity law.)

3. Let \(h: Z\to_M W\) be another arrow. Then

\[ \begin{align*} h\circ_M(g\circ_M f) &= h\circ_M(\mu_Z\circ M(g)\circ f)\\ &= \mu_W\circ \underline{M(h) \circ \mu_Z}\circ M(g)\circ f\\ &= \underline{\mu_W\circ\mu_{MW}}\circ MM(h)\circ M(g)\circ f\\ &= \mu_W\circ M(\mu_W)\circ MM(h)\circ M(g)\circ f\\ &= \mu_W\circ M(\mu_W\circ M(h)\circ g)\circ f\\ &= (h\circ_M g)\circ_M f,\end{align*} \]

by naturality and the associative law of monads.

We see that, indeed, \(C_M\) is a category, where the composition is given by \(\circ_M\), and the identity associated with an object \(X\) is \(\eta_X\). This category is called the Kleisli category of \(M\).

The Kleisli composition is often written an operator like <=<, there’s a flipped form >=> called the “Kleisli fish”.

The Kleisli adjunction

Let’s introduce two functors, \(F_M: C\to C_M\) and \(G_M: C_M\to C\). On objects, \(F_M\) maps each object to itself (recall that the objects of \(C_M\) are precisely the objects of \(C\)). Then \(F_M\) must send a morphism \(f: X\to Y\) to a \(C_M\)-morphism \(F_M(f) : X\to_M Y\), that is, a \(C\)-morphism \(X\to MY\). We can accomplish that by post-composing with an \(\eta\), so we have \(F_M(f) := \eta_Y \circ f: X\to MY\).

\(f: X\to Y\quad\leadsto\quad F_M(f): X\xrightarrow{f} Y\xrightarrow{\eta_Y} MY\)

The other functor, \(G_M\), can’t simply undo this, because there’s no general way to obtain a morphism \(X\to Y\) from \(X\to MY\). We cannot “unpack” monads, but we can “flatten” using \(\mu\). So we define \(G_M(X) = M(X)\) on objects, and \(G_M(f: X\to MY) := \mu_Y \circ M(f)\).

\(f: X\to M(Y)\quad\leadsto\quad G_M(f): M(X)\xrightarrow{M(f)} MM(Y)\xrightarrow{\mu_Y} M(Y)\).

To show that they are adjoint, observe that a \(C\)-morphism \(f: X\to MY\) is simultaneously a morphism \(X\to_M Y\) in \(C_M\), so with our functors, it’s \(F_M(X)\to_M Y\) and \(X\to G_M(Y)\) at the same time. This means that the hom-set bijection is actually an identity.

\[\begin{align*} (\Phi_M)_{XY} : \hom_{C_M}(F_M(X), Y) &\xrightarrow{\cong} \hom_C(X, G(Y))\\ f&\mapsto f \end{align*}\]

What could be more natural than an identity? But not so fast, let’s do this carefully. First, let \(f: T\to X\) be a morphism in \(C\). Then \(\hom_{C_M}(F_M(f), Y)\) is precomposition with \(F_M(f)\), that is, for \(\varphi \in\hom_{C_M}(F_M(X), Y)\), we have

\[ \begin{align*} \hom_{C_M}(F_M(f), Y)(\varphi) &= \varphi\circ_M F_M(f)\\ &= \mu_Y\circ M(\varphi)\circ\eta_X\circ f \\ &= \mu_Y\circ\eta_{MY}\circ\varphi\circ f\\ &= \varphi\circ f\\ &= \hom_C(f, G_M(Y))(\varphi) \end{align*} \]

by naturality of \(\eta\) and the left identity law. For the other variable things are much easier, with \(g: Y\to_M T\), \(\hom_{C_M}(F_M(X), g)(\varphi) = g\circ_M\varphi = \mu_T\circ M(g)\circ\varphi = \hom_C(X, G_M(g))(\varphi)\) for \(\varphi\in\hom_{C_M}(X, Y)\).

Eilenberg-Moore algebras

An \(M\)-algebra (also called an Eilenberg-Moore algebra) is a pair \((X, h)\) of an object \(X\in C\) and an arrow \(h: MX\to X\), such that the following diagrams commute:

\[ \begin{CD} MMX @>{Mh}>> MX \\ @V{\mu_X}VV @V{h}VV\\ MX @>{h}>> X \end{CD}\]

and

\[\begin{CD} X @>{\eta_X}>> MX\\ @| @V{h}VV\\ X @= X \end{CD}\]

A morphism between two \(M\)-algebras, \((X, f: MX\to X)\) and \((Y, g: MY\to Y)\) is a \(C\)-morphism \(h: X\to Y\) which is compatible with the two algebras in the sense that the following diagram commutes:

\[\begin{CD} MX @>{f}>> X\\ @V{Mh}VV @V{h}VV \\ MY @>{g}>> Y \end{CD}\]

(One could also say that a morphism of \(M\)-algebras is a diagram like this.) The morphisms compose in an obvious way and we have the obvious identities, so \(C^M\) is a category, the Eilenberg-Moore category of \(M\).

Some examples

In general, it is not clear for a given object \(X\in C\) that an \(M\)-algebra \(MX\to X\) even exists. For a simple example, if \(M\) is the Maybe monad, then there does not exist an \(M\)-algebra \(h: M(\emptyset)\to\emptyset\), because \(h(\mathsf{Nothing})\in\emptyset\) is a contradiction. It turns out that there is a 1:1 correspondence in this case between \(M\)-algebras \(h: MX\to X\) and pointed sets \((X, x_0\in X)\) via

\[h(x) = \begin{cases}x\text{ if }x\in X\\x_0\text{ if }x = \mathsf{Nothing}\end{cases}.\]

(\(h\) must send everything in \(X\) to itself due to the condition \(h\circ\eta_X = \id_X\).)

If we take the list monad \(L\), then there is a 1:1 correspondence between \(L\)-algebras \(h: LX\to X\) and monoids \((X,\diamond, e)\), by \(h([x_1, \ldots, x_n]) = x_1\diamond\cdots\diamond x_n\).

Interestingly, the list functor is the free functor to monoids, and the maybe functor is the free functor to pointed sets. I’m sure this isn’t an accident.

The Eilenberg-Moore adjunction

Let’s again define a pair of functors, \(F^M: C\to C^M\) and \(G^M: C^M\to C\). The functor \(F^M\) must send an object \(X\in C\) to an \(M\)-algebra, preferably one that is somehow connected with \(X\). But this is difficult in general, as the previous section showed. Fortunately, there is a canonical choice just one more \(M\) away: \(\mu_X : MMX\to MX\) automatically satisfies the conditions for an \(M\)-algebra, so we set \(F^M(X) = (MX, \mu_X)\), the required conditions

\[\begin{CD} MX @>{\eta_{MX}}>> MMX\\ @| @V{\mu_X}VV\\ MX @= MX \end{CD}\]

and

\[\begin{CD} MMMX @>{M\mu_X}>> MMX\\ @V{\mu_{MX}}VV @V{\mu_X}VV\\ MMX @>{\mu_X}>> MX \end{CD}\]

come from the left identity and the associativity condition of the monad \(M\).

A \(C\)-morphism \(f: X\to Y\) must then be sent to a morphism \(F^M(f): (MX, \mu_X)\to(MY, \mu_Y)\), we set \(F^M(f) = M(f)\):

\[ f : X\to Y \quad\leadsto\quad \begin{CD} MMX @>{\mu_X}>> MX\\ @V{MM(f)}VV @V{M(f)}VV\\ MMY @>{\mu_Y}>> MY \end{CD}\]

By naturality of \(\mu\), this diagram always commutes.

The functor \(G^M\) is simpler, an object \((X, f: MX\to X)\) is sent to \(X\), a morphism \(h: (X, f: MX\to X)\to(Y, g:MY\to Y)\) is sent to the arrow \(h: X\to Y\). So \(G^M\) is like a forgetful functor, it simply forgets structure.

We now construct a bijection

\[\Phi^M_{X,(Y,g)}: \hom_{C^M}(F^MX, (Y,g))\xrightarrow{\cong}\hom_C(X,G^M(Y,g))\]

Let \(h: F^MX=(MX, \mu_X) \to (Y,g)\) be a morphism in \(C^M\). We ignore the compatibility condition and just focus on the part \(h: MX\to Y\). By precomposing with \(\eta_X\), we obtain a morphism \(X\to Y\), or \(X\to G^M(Y,g)\), so we can set \(\Phi^M(h) = h\circ\eta_X\).

\[\Phi^M_{X, (Y,g)}\left ( \begin{CD} MMX @>{\mu_X}>> MX\\ @V{Mh}VV @V{h}VV\\ MY @>{g}>> Y \end{CD} \right ) = \begin{CD} X @>{\eta_X}>> MX @>{h}>> Y \end{CD}\]

The inverse is not as easy, because we’re given any old morphism \(k: X\to Y = G^M(Y, g: MY\to Y)\) in \(C\) and have to make up structure to form a commutative diagram. But we can do it:

\[(\Phi^M_{X, (Y,g)})^{-1}(k : X\to Y) = \begin{CD} MMX @>{\mu_X}>> MX\\ @V{MMk}VV @V{Mk}VV\\ MMY @. MY\\ @V{Mg}VV @V{g}VV\\ MY @>{g}>> Y \end{CD} \]

So we’re just setting \((\Phi^M)^{-1}(k) = g\circ M(k)\). If we insert \(\mu_Y : MMY\to MY\) in the empty space in the middle, we see that the upper square commutes by naturality, the lower by the fact that \(g\) is an \(M\)-algebra.

Now I wrote \((\Phi^M)^{-1}\) without checking that this is actually the inverse of \(\Phi^M\). Let’s quickly make up for that. Using the notation of the previous paragraphs,

\[\begin{align*} (\Phi^M)^{-1}(\Phi^M(h: F^M(X)\to (Y, g))) &= (\Phi^M)^{-1}(h\circ\eta_X) \\ &= g\circ M(h\circ\eta_X) \\ &= h\circ\mu_X\circ M(\eta_X) \\ &= h, \end{align*}\]

where we used the fact that \(h\) is a morphism of \(M\)-algebras and the right identity law for monads, and

\[\begin{align*} \Phi^M((\Phi^M)^{-1}(k: X\to Y)) &= \Phi^M(g\circ M(k))\\ &= g\circ M(k)\circ \eta_X\\ &= g\circ \eta_Y\circ k\\ &= k \end{align*}\]

by naturality of \(\eta\) and the fact that \(g\) is an \(M\)-algebra.

The last thing to check is naturality. In the first variable: Let \(f : T\to X\) be a morphism, then for \(h\in\hom_{C^M}(F^MX, (Y,g))\), \(\hom_{C^M}(F^M(f), (Y,g))(h) = h\circ M(f)\), which is sent to \(h\circ M(f)\circ\eta_T = h\circ\eta_X\circ f\) due to naturality, which is just \(\hom_C(f, G^M(Y,g))(\Phi^M(h))\). In the second variable: Let \(f : (Y,g)\to(T,h)\) be a morphism in \(C^M\). Then obviously \(\Phi^M(\hom_{C^M}(F^MX, f)(h)) = f\circ h \circ \eta_X = \hom_C(X, G^M(f))(\Phi^M(h))\) for \(h\in\hom_{C^M}(F^MX, (Y,g))\).

The category of adjunctions that form a given monad

There can be many adjunctions that form a given monad \(M\). Let’s call them \(M\)-adjunctions. To relate them, the elementary idea of category theory tells us: Look for the morphisms!

Following MacLane, let us denote an \(M\)-adjunction by a quadruple \((F, G, \eta, \varepsilon)\), where \(F: C\to D\), \(G: D\to C\) is an adjoint pair of functors with \(M=GF\), \(\eta: \Id\natto GF\) is the unit of the adjunction and the monad, and \(\varepsilon: FG\natto\Id\) is the counit of the adjunction.¹

A morphism of \(M\)-adjunctions \((F_1, G_1, \eta, \varepsilon_1)\to (F_2, G_2, \eta, \varepsilon_2)\) is a functor \(K: D_1\to D_2\), called comparison functor, such that the following diagram commutes:

\[\begin{CD} C @>{F_1}>> D_1 @>{G_1}>> C\\ @| @V{K}VV @|\\ C @>{F_2}>> D_2 @>{G_2}>> C \end{CD}\]

that is, we require \(F_2=K\circ F_1\) and \(G_1 = G_2\circ K\). These morphisms compose in an obvious way, so we obtain the category of \(M\)-adjunctions, \(\mathcal A_M\).

The rest of the article will be dedicated to showing the roles of the Kleisli and the Eilenberg-Moore adjunction in this category.

Theorem:

Let \((M,\eta,\mu)\) be a monad on a category \(C\), and let \(\mathcal A_M\) be the category of \(M\)-adjunctions.

1. The Kleisli adjunction \((F_M, G_M, \eta, \varepsilon_M)\) is an initial object in \(\mathcal A_M\).

2. The Eilenberg-Moore adjunction \((F^M, G^M, \eta, \varepsilon^M)\) is a terminal object in \(\mathcal A_M\).

Proof:

Let \((F, G, \eta, \varepsilon)\) be an \(M\)-adjunction, given by \(\Phi: \hom_C(FX,Y)\cong\hom_D(X,GY)\).

1. We define a functor \(K: C_M\to D\) such that \(K\circ F_M = F\): Since \(F_M\) acts as the identity on objects, we have no choice but to set \(K(X) = F(S)\). On morphisms \(f: X\to M(Y)\), we set \(K(f) = \varepsilon_{FY}\circ F(f)\).

The identity corresponding to an object \(X\) in the Kleisli category is \(\eta_X : X\to_M X\), and \(K(\eta_X) = \varepsilon_{FX}\circ F(\eta_X) = \id_{FX}\) by the triangle equations, so \(K\) preserves identities. If \(f: X\to M(Y)\) and \(g: Y\to M(Z)\) are Kleisli arrows,

\[\begin{align*} K(g\circ_M f) &= \varepsilon_{FZ}\circ F(\mu_Z\circ M(g)\circ f)\\ &= \underline{\varepsilon_{FZ}\circ FG\varepsilon_{FZ}}\circ FGFg \circ Ff\\ &= \varepsilon_{FZ}\circ \underline{\varepsilon_{FGFZ}\circ FGFg} \circ Ff\\ &= \varepsilon_{FZ}\circ Fg\circ \varepsilon_{FY}\circ Ff\\ &= K(g) \circ K(f), \end{align*}\]

by the diagram associated with \(\varepsilon\bullet\varepsilon\) and naturality, so \(K\) is indeed a functor.

Now we’ve already seen \(K(F_M(X)) = F()\) for objects \(X\), and for \(f: X\to Y\), we have \(K(F_M(f)) = K(\eta_X\circ f) = \varepsilon_{FY}\circ F(\eta_X)\circ F(f) = F(f)\) by the triangle equality.

Also, \(G(K(X)) = G(F(X)) = M(X) = G_M(X)\), and if \(f: X\to_M Y\) is a morphism in \(C_M\), \(G(K(f)) = G(\varepsilon_{FY}\circ F(f)) = G(\varepsilon_{FY})\circ M(f) = \mu_Y\circ M(f) = G_M(f)\).

It remains to show that the action of \(K\) on morphisms of \(C_M\) is uniquely determined. Let \(L\) be another functor \(C_M\to D\) such that \(G\circ L = G_M\) and which satisfies \(L(X) = F(X)\) on objects. We’re going to use the interplay between \(\Phi\), \(\eta\) and \(\varepsilon\) now. For a \(C_M\)-morphism \(f: X\to_M Y\), we have \(G(L(f)) = G(K(f))\), hence \(G(L(f))\circ\eta_X = G(K(f))\circ\eta_X\). We can rewrite both sides in terms of \(\Phi\): \(\Phi_{X,FY}(L(f)) = \Phi_{X,FY}(K(f))\), from which follows \(L(f) = K(f)\), because \(\Phi_{X,FY}\) is a bijection. We conclude that \(L=K\).

2. We define a functor \(K: D\to C^M\) such that \(K\circ F = F^M\) and \(G^M\circ K = G\). Since \(K(FX) = F^M(X) = (MX, \mu_X) = (GFX, G(\varepsilon_{FX}))\), an obvious choice is \(K(Y) = (GY, G(\varepsilon_Y))\). This also satisfies \(G^M(KY) = GY\).

For a morphism \(f:X\to Y\) in \(D\), \(K(f)\) is a morphism in the Eilenberg-Moore category between the free \(M\)-algebras \(K(X)\) and \(K(Y)\), but as \(G^M\) is a forgetful functor, the condition \(G^M(Kf) = G(f)\) forces \(K(f) = G(f)\).

\[\begin{CD} MG(X) @>{G(\varepsilon_X)}>> G(X)\\ @V{MG(f)}VV @V{G(f)}VV\\ MG(Y) @>{G(\varepsilon_Y)}>> G(Y) \end{CD} \quad\leadsto\quad \begin{CD} G(X)\\ @V{G(f)}VV \\ G(Y) \end{CD}\]

The remainder of the proof is dedicated to showing that \(K\) is uniquely determined. It is clear that we had no choice in the definition of \(K(Y)\) for objects \(Y\) in the image of \(F\), neither did we have a choice in the definition of \(K(f)\). It is not yet clear that \(K(Y)\) must be \((GY, G(\varepsilon_Y))\) in general. This has to do with the seemingly arbitrary choice of \(\mu_X\) as the “free \(M\)-algebra” \(F^M(X)\). Just because we can’t think of anything else right away doesn’t mean it’s unique.

We proceed in three steps. First, the following diagram is commutative for all \(X\in C\), \(Y\in D\):

\[\begin{CD} \hom_D(FX, Y) @>{\Phi_{XY}}>> \hom_C(X, GY)\\ @V{K_{FX,Y}}VV @|\\ \hom_{C^M}(F^MX, KY) @>{\Phi^M_{X,KY}}>> \hom_C(X, GY) \end{CD}\]

Here, \(K_{FX,Y}\) denotes the action of \(K\) on morphisms, \(\hom_D(FX,Y)\to\hom_{C^M}(KFX, KY),\,f\mapsto K(f)\). Keep in mind that \(KFX = F^MX\) and \(G^MKY = GY\).

To see that this is true, realize that \(\Phi_{XY}(f) = G(f)\circ\eta_X\) and \(\Phi^M_{X,KY}(K(f)) = G^M(K(f))\circ\eta_X = G(f)\circ\eta_X\), too.

Second, recall that the counit \(\varepsilon\) arises as the image of the identity under the inverse adjunction. Thus \(\Phi^M_{GY,KY}(K(\varepsilon_Y)) = \Phi_{GY,Y}(\varepsilon_Y) = \id_{GY} = \id_{G^M(KY)}\), therefore \(K(\varepsilon_Y) = (\Phi^M_{GY,KY})^{-1}(\id_{G^M(KY)}) = \varepsilon^M_{KY}\), for all \(Y\in D\).

Third, \(\varepsilon^M_{(A,h)} = h\) in general, so the structure morphism of \(K(Y)\) is \(\varepsilon^M_{K(Y)} = K(\varepsilon_Y) = G(\varepsilon_Y)\), for all \(Y\in D\).

This concludes the proof.

Additional notes

• Up to isomorphism of functors, the currying adjunction is the only adjunction that stays inside \(\cat{Set}\). This seems to mean that the State monad is the only monad that is formed in this way (up to isomorphism.) I’m going to look into this in another article.