Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 (Category Theory)

For a locally small category $\mathcal{C}$, an object $C$, and a categorical presheaf $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$, the Yoneda lemma gives the natural bijectionIts two maps are explicitlyFor $u:A^{\prime }\to A$, the equation $X(u)X(f)(x)=X(fu)(x)$ proves naturality of $\Psi (x)$. Evaluating it at the identity morphism gives $\Phi \Psi (x)=x$. Conversely, naturality of $\alpha$ at $f:A\to C$ givesso $\Psi \Phi (\alpha )=\alpha$. Evaluation at the identity and transport of an element along a morphism are mutually inverse.

The bijection is natural in both variables: a natural transformation $\theta :X\Rightarrow Y$ sends $\Phi (\alpha )$ to ${\theta }_C(\Phi (\alpha ))$, matching $\Phi (\theta \alpha )$; and $g:C\to C^{\prime }$ givesFor completeness, the covariant Yoneda lemma for $Y:\mathcal{C}\to \mathbf{Set}$ is $\mathrm{Nat}(\mathcal{C}(C,-),Y)\cong Y(C)$, with $\alpha \mapsto {\alpha }_C(1_C)$ and inverse $y\mapsto (f:C\to A\mapsto Y(f)(y))$.

For the categorical presheaf $X$, its category of elements has objects $(C,x)$ with $x\in X(C)$. A morphism $(C,x)\to (D,y)$ is a morphism $f:C\to D$ satisfying $X(f)(y)=x$. Composition in a category is inherited from $\mathcal{C}$: if also $X(g)(z)=y$, then $X(gf)(z)=X(f)X(g)(z)=x$. The identity morphisms are inherited as well. The forgetful functor sends $(C,x)$ to $C$ and $f$ to $f$.

A universal element is a pair $(R,r)$ for which each $x\in X(C)$ is uniquely of the form $X(f)(r)$ for $f:C\to R$. Thus $(R,r)$ is a terminal object of the category of elements, with the variance appropriate to a categorical presheaf.

Given a universal element, defineThe defining uniqueness makes each map a bijection; $X(u){\psi }_C(f)={\psi }_{C^{\prime }}(fu)$ gives naturality for $u:C^{\prime }\to C$. Hence $\psi$ is a natural isomorphism and $X$ is a representable presheaf. Conversely, from a natural isomorphism $\psi :\mathcal{C}(-,R)\cong X$, take $r={\psi }_R(1_R)$. The Yoneda lemma gives ${\psi }_C(f)=X(f)(r)$; its bijectivity makes $(R,r)$ a universal element. Therefore the two descriptions coincide:

For $b:B\to B^{\prime }$ in $\mathcal{B}$, conjugate the induced natural transformation by the chosen representations of a functor:The Yoneda lemma gives a unique morphism $Gb:GB\to G{B}^{\prime }$ whose postcomposition map is ${\tau }_b$. Explicitly,Consequently, for every $f:A\to GB$,This is precisely the required naturality in $B$. The identities ${\tau }_{1_B}=1$ and ${\tau }_{b^{\prime }b}={\tau }_{b^{\prime }}{\tau }_b$ force $G{1}_B=1_{GB}$ and $G(b^{\prime }b)=G{b}^{\prime }Gb$, because the Yoneda embedding is fully faithful. Thus the representing objects extend uniquely to a functor $G:\mathcal{B}\to \mathcal{A}$. Uniqueness follows by the same Yoneda lemma: any candidate satisfying that naturality must induce ${\tau }_b$, so must have exactly the displayed action on morphisms.

Fix the chosen categorical limit object $LA$ and its categorical cone $p_{A,j}:LA\to (DA{)}_j$. For $a:A\to A^{\prime }$, the familyis a categorical cone, by naturality of $Da$. The universal property of $L{A}^{\prime }$ supplies a unique morphismThe identity morphism of $LA$ satisfies the equations for $a=1_A$, so $L{1}_A=1_{LA}$. For composable $a,a^{\prime }$, the equations for $L{a}^{\prime }La$ agree with those for $L(a^{\prime }a)$; uniqueness gives $L(a^{\prime }a)=L{a}^{\prime }La$. Hence the chosen limits define a functor $L:\mathcal{A}\to \mathcal{C}$. This argument also handles an empty indexing category, when each chosen categorical limit is a terminal object.

Equivalently, $LA$ represents the categorical cone functor $C\mapsto \mathrm{Nat}({\Delta }_{J}C,DA)$, and functoriality of chosen representations supplies the same $L$.

Assume $F$ preserves existing categorical limits. For a diagram in a category $E:I\to \mathcal{B}$ with categorical limit cone ${\ell }_i:L\to E_i$, its image $F{\ell }_i$ is a categorical limit cone of $FE$. For any $C\in \mathcal{C}$, a morphism $C\to FL$ is therefore uniquely equivalent to a compatible family of morphisms $C\to F{E}_i$. In the Category of sets, compatible families are exactly the categorical limit of the resulting hom-sets. ThusThe maps are induced by $F{\ell }_i$, so $\mathcal{C}(C,F-)$ preserves the given limit. Applying this to every existing categorical limit proves the implication, with no completeness hypothesis on $\mathcal{B}$.

Conversely, assume that every hom-set functor $\mathcal{C}(C,F-)$ preserves existing categorical limits. For a categorical limit cone ${\ell }_i:L\to E_i$ in $\mathcal{B}$, a categorical cone $q_i:C\to F{E}_i$ determines an element of ${\lim }_i\mathcal{C}(C,F{E}_i)$. The assumed bijectiongives one and only one such $u$ for every $C$ and every categorical cone. That is the universal property of the image cone itself. Therefore the hom-set tests detect limit preservation:This proof works for all diagram sizes for which the relevant categorical limits and compatible-family sets are under consideration.

The representing object in the functor category $[J,\mathcal{C}]$ is the constant diagram in a category ${\Delta }_{J}C$. If $p_j:{\lim }_j{E}_j\to E_j$ is the chosen categorical limit cone, the natural bijection isA natural transformation from the constant diagram in a category is precisely a categorical cone with vertex $C$, and its inverse is the unique mediating morphism from the universal property of the categorical limit. For $\alpha :E\Rightarrow E^{\prime }$, the defining equations for $\lim \alpha$ show that postcomposition by $\lim \alpha$ corresponds to postcomposition by $\alpha$ on the right. This establishes naturality in $E$ and proves representability by the constant diagram.

For every $C$, the preceding part identifies the composite $\mathcal{C}(C,{\lim }_J-)$ with the covariant representable functor $[J,\mathcal{C}]({\Delta }_{J}C,-)$. A covariant representable functor preserves every existing categorical limit, since a morphism into a categorical limit is the same as a compatible family of morphisms into the diagram objects. The limit functor preserves limits. The hom-set detection of categorical limits givesIn particular, it preserves all small categorical limits. They exist in $[J,\mathcal{C}]$: compute each one at every $j$ using completeness of $\mathcal{C}$, and use the uniqueness of the pointwise mediating morphisms to obtain its functor structure and universal natural transformations. Another expression of the same result is the adjunction ${\Delta }_J⊣{\lim }_J$.

Regard the bifunctor as $D^{(2)}:K\to [J,\mathcal{C}]$. Its pointwise categorical limit $E={\lim }_k{D}^{(2)}k$ satisfieswith its action on morphisms obtained by the construction in part (a). Applying the limit-preservation result from part (d) givesThe comparison isomorphism is canonical relative to the chosen categorical limit cones: it is the unique morphism identifying all the projections to $D(j,k)$. This proves commutation of iterated categorical limits, including an empty $J$ or $K$. Equivalently, both iterated constructions have the universal property of the categorical limit of $D$ on $J\times K$.

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.

Suppose $X\cong \mathcal{C}(-,R)$. As a functor on the opposite category, a representable presheaf preserves every existing categorical limit: a colimit in $\mathcal{C}$ is defined by the bijection between morphisms from its vertex into $R$ and compatible families of morphisms from its diagram objects into $R$. This remains valid for a possibly large diagram in a category whenever that colimit exists and the compatible-family collection is the corresponding set.

The category of elements has a terminal object $(R,r)$, where $r$ is the image of $1_R$ under the representation. For each $(C,x)$ let $u_{C,x}:C\to R$ be its unique morphism to $(R,r)$. These form a cocone for the forgetful functor $G$. Any competing cocone $v_{C,x}:C\to A$ satisfiesThe component at $(R,r)$ uniquely determines the mediating morphism. Thus the representing object is the colimit of the elements projection:

Let $u_{C,x}:C\to R$ be a colimit cocone for $G:\mathrm{Elts}(X)\to \mathcal{C}$. Here the preservation hypothesis must include the categorical limit of $G^{\mathrm{op}}$ in ${\mathcal{C}}^{\mathrm{op}}$; this indexing category can be large. For a morphism $f:(C,x)\to (D,y)$ of the category of elements, the defining equation $X(f)(y)=x$ says that the family of distinguished elements $(x{)}_{(C,x)}$ is compatible. Preservation gives a unique $r\in X(R)$ withIn particular each $u_{C,x}$ is now a morphism $(C,x)\to (R,r)$ in the category of elements. Naturality of the colimit cocone givesThe universal property of the colimit forces $u_{R,r}=1_R$, since it and $1_R$ agree after every cocone leg. If $f:C\to R$ also satisfies $X(f)(r)=x$, it is a morphism $(C,x)\to (R,r)$, and cocone compatibility now yields $u_{C,x}=u_{R,r}f=f$. Thus $(R,r)$ is a universal element. By the preceding representability criterion, the presheaf is represented by the colimit object:The size qualification matters. If preservation means only small categorical limits, the implication as printed is false for general locally small categories. For an explicit counterexample, let $\mathcal{C}$ be the ordered category of all ordinals with an extra greatest object $\infty$. Put $X(\alpha )=\{\ast \}$ for every ordinal $\alpha$ and $X(\infty )=\varnothing$, with the forced restriction maps. Every small colimit in $\mathcal{C}$ is the supremum of its object values: it is an ordinal unless the diagram contains $\infty$. Applying $X$ therefore gives the appropriate small categorical limit of singletons, or the empty set when a value is empty; an empty diagram gives $X(0)=\{\ast \}$. Hence $X$ preserves all small categorical limits.

Its category of elements consists of all ordinals, whose projection has the large colimit $\infty$ in $\mathcal{C}$. Nevertheless, no ordinal represents $X$, because its representable vanishes on larger ordinals; $\infty$ does not represent it either, since $\mathcal{C}(\infty ,\infty )$ is nonempty while $X(\infty )$ is empty. The valid proof consequently uses preservation of the displayed possibly large limit, or a smallness hypothesis making that diagram small.

The right-adjoint criterion using comma-category colimits is as follows. Under the given colimit-preservation hypothesis, with preservation including the possibly large colimits below, $F:\mathcal{X}\to \mathcal{A}$ has a right adjoint exactly when, for every $A$, the projectionhas a colimit in $\mathcal{X}$. The right adjoint is obtained from these comma-category colimits. Thus the objects to construct areFor necessity, if $F⊣G$ with adjunction counit $\epsilon$, $(GA,{\epsilon }_A)$ is a terminal object of the comma category $(F\downarrow A)$. Its unique incoming morphisms give a colimit cocone for $U_A$, exactly as for the elements projection in the preceding part.

For sufficiency, choose such a colimit $L$ with legs $u_{X,f}:X\to L$. The morphisms $f:FX\to A$ are a cocone on $F{U}_A$. Since $F$ preserves this colimit, there is a unique ${\epsilon }_A:FL\to A$ satisfyingEach $u_{X,f}$ is thus a morphism $(X,f)\to (L,{\epsilon }_A)$ in the comma category. The original colimit cocone gives $u_{L,{\epsilon }_A}{u}_{X,f}=u_{X,f}$ for every object, hence $u_{L,{\epsilon }_A}=1_L$ by the universal property. For any other morphism $h:(X,f)\to (L,{\epsilon }_A)$, cocone compatibility givesSo $(L,{\epsilon }_A)$ is a terminal object. Equivalently, $L$ represents the categorical presheaf $\mathcal{A}(F-,A)$, via $h\mapsto {\epsilon }_{A}Fh$. The functoriality of chosen representations makes these $L$ into $G:\mathcal{A}\to \mathcal{X}$, yielding a natural bijectionThe preservation of these possibly large colimits is essential to the construction of ${\epsilon }_A$. Under a small-only interpretation, the preceding ordinal counterexample also disproves the unqualified converse here. Take $F=X^{\mathrm{op}}:\mathcal{C}\to {\mathbf{Set}}^{\mathrm{op}}$. It preserves all small colimits. For a nonempty set $A$, the comma category $(F\downarrow A)$ has one object over each ordinal and none over $\infty$, so its projection has colimit $\infty$. For $A=\varnothing$, the projection is the identity of $\mathcal{C}$, again with colimit $\infty$. Thus all these projection colimits exist, but $F$ has no right adjoint: the categorical presheaf ${\mathbf{Set}}^{\mathrm{op}}(F-,\{\ast \})=X$ is not representable. This makes the large-preservation qualification substantive.

The general adjoint functor theorem has the following limit form. Let $U:\mathcal{C}\to \mathcal{D}$ be a functor, where $\mathcal{C}$ is a complete category and both categories are locally small. Then $U$ has a left adjoint exactly when it preserves small limits and satisfies the solution-set condition.

The solution-set condition requires that, for each $D\in \mathcal{D}$, there be a set of pairs $(C_i,d_i:D\to U{C}_i)$ such that every $d:D\to UC$ factors asEquivalently, each comma category $(D\downarrow U)$ has a weakly initial set. All completeness and preservation requirements here concern small categorical limits. Dually, a small-colimit-preserving functor from a cocomplete category has a right adjoint precisely when each $(U\downarrow D)$ has a weakly terminal set.

The Special adjoint functor theorem says: if $\mathcal{C}$ is a locally small category, a complete category, a well-powered category, and has a small cogenerating family, then a functor $U:\mathcal{C}\to \mathcal{D}$ into any locally small category has a left adjoint if and only if it preserves small categorical limits.

A small cogenerating family $(Q_i{)}_{i\in I}$, with $I$ a set, distinguishes unequal parallel morphisms by postcomposition: for $f\ne g:A\to B$, some $q:B\to Q_i$ has $qf\ne qg$. Being well-powered means that the subobjects of each object form a set up to the usual equivalence of monomorphisms. These hypotheses make the solution-set condition automatic.

The dual formulation uses a cocomplete category, being a well-copowered category and having a small generating family, and concludes that every small-colimit-preserving functor has a right adjoint.

It is enough to prove the solution-set condition and then apply the general adjoint functor theorem. Fix $D\in \mathcal{D}$ and a morphism $d:D\to UC$. We will factor it through one object in a set depending only on $D$.

First construct a minimal supported subobject. Among the subobjects $m:M\hookrightarrow C$ for which $d=U(m)d_M$ for some $d_M:D\to UM$, include $1_C$ and take their intersection $m_0:C_0\hookrightarrow C$. This is a small intersection, by well-poweredness. It exists by completeness as the categorical limit of the diagram consisting of these monomorphisms into $C$. Its map to $C$ is a monomorphism: two maps with the same composite to $C$ have equal projections to every $M$, since each $m$ is monic, and are then equal by the categorical limit property. One can equivalently construct these intersections by pullbacks in a category and small products in a category, using stability of monomorphisms under pullback in a category.

Choose the factorizations $d_M$. They form a compatible categorical cone into the image diagram under $U$, all with common composite $d$ to $UC$. Preservation of small categorical limits yields $d_0:D\to U{C}_0$ with $U(m_0)d_0=d$. If $n:N\hookrightarrow C_0$ is another subobject through which $d_0$ factors after applying $U$, then $m_{0}n$ is among the original supported subobjects. The intersection property gives $r:C_0\to N$ with $m_{0}nr=m_0$. Since $m_0$ is monic, $nr=1_{C_0}$; since $n$ is monic, also $rn=1_N$. Thus every supported subobject of $C_0$ is invertible.

For each member $Q_i$ of the small cogenerating family, considerThis map is injective. If $U(q)d_0=U(q^{\prime })d_0$, preservation of the equalizer of $q,q^{\prime }$ makes $d_0$ factor through the image of that equalizer. Minimality makes its inclusion an isomorphism, so $q=q^{\prime }$.

Write $R_i=\mathcal{D}(D,U{Q}_i)$; it is a set by local smallness of $\mathcal{D}$. Let $S_i\subseteq R_i$ be the image of ${\theta }_i$. For each $s\in S_i$, there is exactly one corresponding $q_{i,s}:C_0\to Q_i$. These maps define the evaluation embedding into cogenerator productsIt is a monomorphism: if $eg=eh$ and $g\ne h$, cogeneration supplies some $q:C_0\to Q_i$ distinguishing $g,h$; that $q$ is one of the projections of $e$, a contradiction. The product is small. It is important to use the subfamilies $S_i$, since some missing coordinate in $R_i$ need not correspond to a morphism out of $C_0$.

There are only a set of possible families $(S_i\subseteq R_i{)}_{i\in I}$. For each such family form $P_S$, choose a set of representatives $N\hookrightarrow P_S$ of its subobjects, and take all pairs $(N,d_N:D\to UN)$. Local smallness of $\mathcal{D}$ and well-poweredness make their union a set. In the case just constructed, $e$ identifies $C_0$ with one chosen representative $N$. Transporting $d_0$ to $UN$ and composing the inverse identification with $m_0$ gives a factorization of the original $d$ through that pair. Therefore these pairs form a weakly initial set in $(D\downarrow U)$.

The general adjoint functor theorem now supplies a left adjoint to $U$. Conversely, a right adjoint preserves small categorical limits, either by the adjunction hom-set bijections and the hom-set detection of categorical limits, or directly from their universal properties. Hence the stated special theorem is proved in both directions.

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 $h\mapsto {\epsilon }_{C^{\prime }}K(h){\epsilon }_C^{-1}$.

If $B$ belongs to the full subcategory $\mathcal{C}$, the adjunction and fullness of its inclusion give, for every $B^{\prime }\in \mathcal{B}$,Therefore membership implies the hom-set condition. In addition, $J$ is fully faithful, so the adjunction counit is invertible by the preceding criterion. The triangular equation $J{\epsilon }_B{\rho }_B=1_B$ then shows that ${\rho }_B$ is an isomorphism. As in the printed inclusion notation, $J$ is suppressed when writing $KB$ as an object of $\mathcal{B}$.

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.

If ${\rho }_B:B\to KB$ is an isomorphism, then $B$ is isomorphic in $\mathcal{B}$ to an object of $\mathcal{C}$. Since $\mathcal{C}$ is a replete subcategory, it contains $B$. Thus invertibility of the unit implies membership, completing the cycle of implications:

Let $J:\mathcal{C}\hookrightarrow \mathcal{B}$ exhibit a reflective subcategory, with reflector $K$, and let $E:I\to \mathcal{C}$ be a small diagram in a category. Completeness of $\mathcal{B}$ gives a categorical limit $L$ of $JE$. For every $B^{\prime }\in \mathcal{B}$, the universal property of this categorical limit and the reflection adjunction giveThe composite is precomposition with ${\rho }_{B^{\prime }}$, so $L$ satisfies the hom-set condition from the preceding part. Its proof of invertibility of ${\rho }_L$ did not require repleteness. Thus $L\cong JKL$ whether or not the chosen full reflective subcategory is replete.

Transport the ambient limit cone along ${\rho }_L^{-1}:JKL\to L$. Its legs lie in the full subcategory, and their ambient universal property, restricted to objects of $\mathcal{C}$, is exactly the internal categorical limit property. Hence every small diagram in the reflective subcategory has a limit:For a replete subcategory, the ambient limit object $L$ itself belongs to $\mathcal{C}$. This argument includes the empty diagram and requires no limit-preservation hypothesis on the reflector.

For the monad $\mathbb{T}=(T,\eta ,\mu )$ define the free algebra functor byThe unit and associativity identities of the monad make $(TX,{\mu }_X)$ an algebra for a monad, and naturality of $\mu$ makes $Tf$ a morphism of algebras for a monad. Let $G^{\mathbb{T}}$ be the forgetful functor. For $(A,a)$ in the Eilenberg-Moore category, setThe inverse candidate is a monad algebra morphism, sinceThe two composites are identities:These use respectively the unit law for a monad algebra, the algebra-morphism equation, and a unit identity of the monad. The formulas commute with precomposition in $X$ and postcomposition by monad algebra morphisms, so they form a natural bijection. Therefore the free-algebra functor is left adjoint to forgetting:Its adjunction unit is ${\eta }_X$, and its adjunction counit at $(A,a)$ is $a:(TA,{\mu }_A)\to (A,a)$.

Write $F:\mathcal{X}\to \mathcal{A}$ and $G:\mathcal{A}\to \mathcal{X}$, with adjunction counit $\epsilon$. The induced monad has $T=GF$ and $\mu =G\epsilon F$. Define the Eilenberg-Moore comparison functor byThe action $a_A=G{\epsilon }_A:TGA\to GA$ satisfies $a_A{\eta }_{GA}=1_{GA}$ by a triangular equation. Naturality of $\epsilon$ at ${\epsilon }_A:FGA\to A$ givesApplying $G$ yields $a_{A}T(a_A)=a_A{\mu }_{GA}$, the other algebra identity. For $f:A\to A^{\prime }$, naturality givesso $Gf$ is a monad algebra morphism. Identities and compositions are respected because $G$ is a functor. Finally,and the same equalities hold on morphisms. Thus both comparison identities hold, strictly when $\mathbb{T}$ is the specified induced monad:

If the multiplication in the unit and multiplication of a monad $\mu$ is invertible, the two unit identities of the monadmake both $T\eta$ and $\eta T$ its inverse. Hence idempotence implies equality of the two unit insertions:This is the first implication in the characterization of an idempotent monad.

Assume $T\eta =\eta T$, and let $(A,a)$ be an algebra for a monad. Its unit identity gives $a{\eta }_A=1_A$. Naturality of $\eta$ at $a:TA\to A$, followed by the assumed equality, givesThus every algebra action is invertible, with its inverse prescribed by the unit:

Suppose every algebra action is an isomorphism. Its inverse is ${\eta }_A$, by the unit law for a monad algebra. For any underlying morphism $f:A\to B$ between two algebras for a monad, naturality of $\eta$ givesTherefore every underlying morphism is automatically a monad algebra morphism. The forgetful functor is always faithful, because a monad algebra morphism has no data beyond its underlying morphism, and it is now full as well. Hence invertible algebra actions imply

If $G^{\mathbb{T}}$ is fully faithful, the criterion proved for a right adjoint says that the adjunction counit of $F^{\mathbb{T}}⊣G^{\mathbb{T}}$ is a natural isomorphism. Its component at $(A,a)$ is precisely the algebra action $a$. In particular, its component at the free algebra for a monad $(TX,{\mu }_X)$ has underlying morphism ${\mu }_X:T^{2}X\to TX$. Thus every ${\mu }_X$ is an isomorphism, and the natural transformation $\mu$ is invertible. This closes the cycle and proves all four conditions equivalent: