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Fix a monad on $\mathcal{C}$. Objects are adjunctions with left-hand category $\mathcal{C}$ whose induced monads are identified with the fixed monad. A morphism between two such adjunctions is a functor between their right-hand categories commuting with both adjoints and respecting the specified units and counits. Strict identifications give a category; coherent identifications give the analogous universal property up to compatible natural isomorphism.

Category theory studies mathematical structures through the morphisms between them. A category specifies which objects of a category and arrows are allowed; a functor translates these objects and arrows into another category; a natural transformation compares two such translations coherently. For example, the category of groups records both groups and group homomorphisms, so it remembers which maps respect multiplication.

The usefulness appears in constructions we already know. A Cartesian product combines two sets, while a direct product of groups combines two groups. Both satisfy the same universal property of a categorical product: any pair of maps into the factors comes from exactly one map into the product. This gives a reusable way to construct maps and establish isomorphisms, without depending on a particular choice of coordinates or element labels. In a partially ordered set regarded as a category, the corresponding categorical product is a greatest lower bound. The shared definition exposes a connection between pairing data and finding the best common bound.

Start with the computed examples under category, functor, natural transformation, and product in a category. Their arrow equations are practical tests: do both routes in a commutative diagram produce the same result? Later, the same language organizes tensor products, adjunctions, and the fundamental group of a topological space. Emily Riehl's textbook develops these basic notions before using them to compare constructions across different subjects. ^[ref]

In their 1945 paper, Samuel Eilenberg and Saunders Mac Lane set out the notions of category, functor, and natural transformation. The introduction uses the relation between a vector space and its dual space to ask which comparisons are natural, setting up a precise language for compatibility with maps. ^[ref]

For small $J,K$ and a functor $D:J\times K\to \mathcal{C}$ into a complete category, the two iterated categorical limits are canonically isomorphic. They have the same projections to every $D(j,k)$ and the same universal property. The result follows also from preservation of categorical limits by the limit functor.

A mathematical definition fixes the meaning of a term or symbol using previously specified concepts. A definition by a universal property can still require a mathematical proof that an object with that property exists. For example, the definition of a product in a category specifies its mapping behavior, while existence is a further question about that category.

The free object on generators $x_1,\dots ,x_n$ has the universal property that an arbitrary assignment of these generators to an Abelian lattice-ordered group extends uniquely to a lattice-ordered group homomorphism. It can be constructed from terms using addition, negation, zero, join and meet, modulo identities valid in every Abelian lattice-ordered group. Its underlying group need not be finitely generated even when there are only two lattice generators.

A categorical cone $L\to E_i$ is a categorical limit precisely when $\mathcal{C}(C,L)\to {\lim }_i\mathcal{C}(C,E_i)$ is a bijection for every object $C$. This is exactly the universal property expressed as a test on hom-sets. Consequently a functor preserves existing categorical limits if every composite with a covariant representable functor does.

In a commutative diagram of two kernel rows, a monic arrow between the final objects makes the square between their kernels and middle objects a pullback in a category. The kernel universal property factors any categorical cone uniquely. Conversely, if the square between middle and final objects is a pullback, the induced arrow between their kernels in a category is an isomorphism, since both kernels have the same universal property after pullback.

If $G:\mathcal{C}\to \mathcal{D}$ preserves small categorical limits, the projection $(X\downarrow G)\to \mathcal{C}$ creates the small categorical limits available in $\mathcal{C}$. A categorical cone of arrows $X\to G{A}_j$ induces a unique arrow $X\to G(\lim A_j)$ by preservation; this equips the underlying categorical limit with its comma category structure. The universal property proves that every categorical cone factorization preserves that structure.

A specification of the meaning of a mathematical term or notation. A structural definition specifies conditions, such as the group axioms; a definition by a universal property characterizes an object by its arrows. Such a characterization does not by itself prove that the object exists. For example, existence of an exponential object is an additional property of a category.

Use the explicit construction of small limits from products and equalizers from part (i), here with finite indexing sets. If $F$ preserves finite products in a category and equalizers, then $F(P)$ and $F(Q)$, with their image projections, are the products of the objects $FD(j)$ and of the target objects indexed by arrows of $J$.

The image maps $F(a),F(b)$ have componentsAlso $F(e):F(L)\to F(P)$ is their equalizer. Therefore the same universal property proof makes $F(L)$, with projections $F(p_j)$, a categorical limit of $FD$. Any other chosen limit of $D$ is uniquely isomorphic to this construction, and the functor carries that isomorphism to an isomorphism. Consequently $F$ preserves every finite limit. Preservation of finite products includes the empty product, so the terminal object is also preserved.

Let $D:J\to [\mathcal{C},\mathcal{S}]$ be a small diagram in a category. Since $\mathcal{S}$ is a complete category, choose at each $C\in \mathcal{C}$ a categorical limitFor $f:C\to C^{\prime }$, the arrows $D_j(f)p_{j,C}$ form a cone over a diagram at $C^{\prime }$, because the diagram arrows $D_j\to D_k$ are natural transformations. There is therefore a unique arrow $L(f):L(C)\to L(C^{\prime })$ withThe universal property immediately gives $L(1_C)=1_{L(C)}$ and $L(gf)=L(g)L(f)$: both sides have the same composites with every limit projection. Thus $L$ is a functor, and the displayed equations make the $p_j$ natural transformations.

Given any categorical cone $q_j:X\to D_j$ in the functor category, its components induce unique $q_C:X(C)\to L(C)$. For $f:C\to C^{\prime }$, compare $L(f)q_C$ and $q_{C^{\prime }}X(f)$ after each $p_{j,C^{\prime }}$; naturality of $q_j$ makes their composites equal. Hence $q$ is a natural transformation, and uniqueness is componentwise. This proves that $L$ is a categorical limit in $[\mathcal{C},\mathcal{S}]$.

The functor category is complete, and every evaluation functor preserves limits, since evaluating the constructed limit at $C$ gives exactly the chosen limit in $\mathcal{S}$. This is the pointwise limits in a functor category construction; it also covers the empty diagram and its terminal object.

Write the Yoneda embedding as $h:\mathcal{C}\to \hat{\mathcal{C}}$. For any finite product in a category, its universal property givesThese bijections are natural, so $h$ preserves finite products. Now use the exponential in a presheaf category, followed by the Yoneda lemma and the exponential object property in $\mathcal{C}$:Every step is natural in $D$, so these component bijections form a natural isomorphismUnder this isomorphism, the image of the evaluation $B^A\times A\to B$ is the presheaf evaluation: both correspond to the same identity of the exponential under the transpose bijection. Thus the Yoneda embedding preserves exponentials, including their evaluation maps.

If $F⊣U$, let $1$ denote a singleton set. The adjunction gives the natural bijectionsThus $U$ is represented by $F1$, proving $A\Rightarrow R$.

If $U\cong \mathcal{C}(R,-)$ is a covariant representable functor, a morphism $R\to {\lim }_j{C}_j$ is exactly a compatible family of morphisms $R\to C_j$. Consequently the comparison mapis a bijection for every existing small categorical limit. This proves $R\Rightarrow L$ by the universal property of a limit. In particular,

The general adjoint functor theorem says that, for a functor $U:\mathcal{C}\to \mathcal{D}$ with $\mathcal{C}$ a complete category and both categories locally small, $U$ has a left adjoint exactly when it preserves small categorical limits and satisfies the solution-set condition. The latter means that for each $D\in \mathcal{D}$ there is a set of arrows $f_i:D\to U(C_i)$ such that every $f:D\to U(C)$ factors asEquivalently, $(D\downarrow U)$ has a weakly initial set.

The limit form of the special adjoint functor theorem says that, if in addition $\mathcal{C}$ is well-powered and has a small cogenerating family, then $U$ has a left adjoint exactly when it preserves small limits. These are the limit/left-adjoint forms; their duals interchange limits with colimits, left with right adjoints, and cogenerators with generators.

For the first route, let $\mathcal{E}$ be a complete category that is locally small and has a weakly initial set $(W_i)$. Put $P={\prod }_i{W}_i$. It is weakly initial: for any $X$, choose an arrow $W_i\to X$ and precompose with the corresponding product projection. The set $\mathcal{E}(P,P)$ indexes a simultaneous equalizerIt exists by completeness, for example as the equalizer of the two maps $P\rightrightarrows {\prod }_{u:P\to P}P$ with components $u$ and $1_P$. Since $E\to P\to X$ exists for each $X$, $E$ is also weakly initial.

Choose $r:P\to E$. Applying the defining equation to $er:P\to P$ gives $ere=e$, whence $re=1_E$ because $e$ is a monomorphism. If $v:E\to E$ is any endomorphism, apply it to $evr:P\to P$ to obtain $evre=e$, hence $ev=e$ and $v=1_E$. Thus $E$ has only its identity endomorphism.

For two arrows $a,b:E\rightrightarrows X$, take their equalizer $j:Y\to E$. Weak initiality of $E$ gives $t:E\to Y$. Since $jt$ is an endomorphism of $E$, $jt=1_E$. The monic arrow $j$ with this right inverse is invertible; therefore $a=b$. Existence of an arrow $E\to X$ and this uniqueness show that $E$ is initial.

For the general adjoint functor theorem, limit preservation makes each comma category $(D\downarrow U)$ complete, and the solution-set condition makes it weakly initially generated by a set. The lemma provides an initial object $(FD,{\eta }_D)$. For $v:D\to D^{\prime }$, the universal property uniquely defines $Fv:FD\to F{D}^{\prime }$ byUniqueness gives the functor laws. The same property gives natural bijectionsso $F⊣U$. Conversely a right adjoint preserves limits, by applying its hom-set bijections to cones, and its unit $(FD,{\eta }_D)$ is already an initial object of each comma category, hence a singleton solution set. This proves necessity as well as sufficiency.

For the second route, assume the general adjoint functor theorem and the additional hypotheses of the Special adjoint functor theorem. Let $(Q_i{)}_{i\in I}$ be a small cogenerating family. Fix $D\in \mathcal{D}$ and an arrow $f:D\to UC$. A limit-preserving functor preserves monomorphisms, by the self-pullback criterion. Intersect the set of all subobjects $B\hookrightarrow C$ supporting $f$. Preservation of this categorical limit gives a minimal supporting pair $(C_0,f_0:D\to U{C}_0)$ and a monomorphism $m_0:C_0\to C$ with $U(m_0)f_0=f$.

If $g,h:C_0\rightrightarrows Q_i$ have $Ug{f}_0=Uh{f}_0$, their equalizer supports $f_0$ and must be invertible by minimality. ThusThe evaluation map into the product of all these cogenerator targets is monic, because the family coseparates arrows. Its coordinates inject into the fixed setFor every subset $J\subseteq B_D$, form the product $P_J$ whose coordinate $(i,z)$ has target $Q_i$. Each minimal $C_0$ is isomorphic to a subobject of one such product, using precisely its realized coordinates. There is a set of subsets $J$; the well-powered category condition gives a set of representative subobjects of each product. For each representative $B$, include every arrow $D\to UB$, a set by local smallness of $\mathcal{D}$.

This is a solution-set condition: the minimal pair is isomorphic to one of these representatives, and its inclusion into $C$ supplies the required factorization of $f$. The general adjoint functor theorem now gives the left adjoint. This is the cogenerator bound for comma-category solution sets proof of the special theorem. Necessity follows again because a right adjoint preserves limits. Both alternative proof routes are therefore supplied.

First prove the exact lifting statement for the forgetful functor $U^T:{\mathcal{C}}^T\to \mathcal{C}$. Take any diagram in a category of algebras for a monad $(A_j,a_j)$ whose underlying diagram has a chosen categorical limit $p_j:L\to A_j$. The arrows $a_{j}T(p_j):TL\to A_j$ form a categorical cone, since every diagram arrow is a morphism of algebras for a monad. Thus the universal property gives a unique arrow $a:TL\to L$ satisfyingThe projections are jointly monic: two arrows into $L$ with the same composites with every $p_j$ are equal. Therefore the algebra identities can be checked after these projections. For the unit,For multiplication,The algebra law for $a_j$ makes the right-hand sides equal. Thus $(L,a)$ is an algebra for a monad, and each $p_j$ is a morphism of algebras for a monad. Its action is unique with this property.

Given an algebra cone $q_j:(B,b)\to (A_j,a_j)$, let $q:B\to L$ be its unique underlying mediating map. Thenso $qb=aT(q)$ by joint monicity. Hence $q$ is an algebra map, proving the lifted cone is limiting. This establishes that the monad algebra forgetful functor creates limits, without any assumption that $T$ preserves limits.

For a monadic adjunction, the comparison satisfies $U=U^{T}K$. If $K$ is an isomorphism of categories over the base, the preceding unique lift transports back literally, so $U$ is a limit-creating functor. If monadic means that $K$ is only an equivalence of categories, the precise conclusion is creation of limits up to isomorphism: the lifted algebra is isomorphic to an object $K(D)$, and its limiting cone transports along that isomorphism. Full faithfulness supplies the arrows and their uniqueness.

This distinction concerns prescribed underlying objects, not existence of limits. Literal creation does not follow from an arbitrary equivalence alone. For example, the inclusion of the one-object category into the two-object indiscrete category, choosing one of its objects, is an equivalence of categories and is monadic under the equivalence convention. Either object of the target is terminal, but the other chosen terminal object has no literal object preimage. Thus the strict reading of the requested assertion needs the strict comparison convention; the equivalence-invariant statement is proved above.

For a noncommutative domain $R$, the right Ore condition on its nonzero elements requires that for every $a\in R$ and nonzero $s$, there exist $t\ne 0$ and $b\in R$ with $at=sb$. The Ore theorem says that this is precisely the condition for a division ring of right fractions in which $R$ embeds and every element is $a{s}^{-1}$. The left-handed condition similarly gives left fractions.

For completeness, a right Noetherian domain satisfies the condition. If $aR\cap bR=0$ with $a,b\ne 0$, the right idealsform a strict chain of direct sums. Indeed, in a relation ${\sum }_{i=0}^n{a}^{i}b{r}_i=0$, the first term lies in $bR$ and the rest in $aR$. Their zero intersection makes $b{r}_0=0$, hence $r_0=0$; cancel $a$ and repeat. Every new summand is nonzero in a noncommutative domain, contradicting the ascending chain condition. Thus $au=bv\ne 0$ exists, with $u,v\ne 0$, giving right common multiples. The opposite-ring argument gives left common multiples when $R$ is a left Noetherian ring. This proves right Noetherian domains satisfy the Ore condition.

A sketch of the fraction construction also explains the theorem. Pairs $(a,s)$ represent $a{s}^{-1}$. They represent the same fraction when a common right denominator $su=tv\ne 0$ also satisfies $au=bv$. Addition uses $(au+bv)(su{)}^{-1}$. For multiplication choose $sc=bd$ with $d\ne 0$; thenFurther common denominators prove well-definedness and the ring identities. The original map is injective because all nonzero denominators are regular; a zero fraction numerator would satisfy $au=0$ for $u\ne 0$. Every nonzero fraction has inverse $s{a}^{-1}$, so the ring is a division ring. This construction has the universal property of inverting all nonzero elements.

Apply this to the two-sided Noetherian noncommutative domain $kG$. Its left Ore property expresses any right fraction as a left one: choose $u\ne 0$ with $ua=bs$, yielding $a{s}^{-1}=u^{-1}b$. Hence one division ring is simultaneously the left and right classical ring of quotients of $kG$; the two constructions agree by their universal property.

Here is a precise base change of a pullback square statement. Suppose $A=B{\times }_{D}C$ is a pullback in a category and $t:D^{\prime }\to D$ is a morphism. Assume the pullbacks in a categoryexist, where $A\to D$ is the common composite. The induced square with vertices $A^{\prime },B^{\prime },C^{\prime },D^{\prime }$ is a pullback in a category; equivalently,Indeed, compatible morphisms $T\to B^{\prime }$ and $T\to C^{\prime }$ give morphisms $b:T\to B$, $c:T\to C$, and a common $v:T\to D^{\prime }$, with the composites of $b,c$ to $D$ both equal to $tv$. The original pullback in a category gives a unique morphism $h:T\to A$ with projections $b,c$. Its composite to $D$ is $tv$, so the defining pullback in a category for $A^{\prime }$ gives a unique morphism $T\to A^{\prime }$ above $h,v$. Its maps to $B^{\prime }$ and $C^{\prime }$ are the prescribed ones by uniqueness in those pullbacks in a category. Conversely, these projections recover $b,c,v$, so every possible factorization is the same. This proves the universal property without requiring arbitrary categorical limits in the ambient category.

Let $1$ be a terminal object. The pullback in a category of $X\to 1\leftarrow Y$ has precisely the universal property of the binary product in a category $X\times Y$: every pair of morphisms to $X,Y$ is compatible over $1$. Repeated binary products in a category, together with $1$ as the empty product, give all finite products in a category.

For $f,g:X\rightrightarrows Y$, form the pullback in a category of the categorical diagonal ${\Delta }_Y:Y\to Y\times Y$ along $⟨f,g⟩:X\to Y\times Y$. Its projection $e:E\to X$ satisfies $fe=ge$. If $h:T\to X$ satisfies $fh=gh$, the pair $h,fh$ gives a unique morphism into this pullback in a category. Conversely any such morphism must project to that pair. Hence $e$ is the equalizer of $f,g$. The preceding construction now gives all finite limits.

For a functor $D:\mathcal{J}\to \mathcal{C}$ with finite indexing category, form the finite products in a categoryDefine $s,t:P\rightrightarrows Q$ by requiring the coordinate indexed by $u:i\to j$ to be $D(u){\pi }_i$ for $s$ and ${\pi }_j$ for $t$. Let $e:L\to P$ be their equalizer, and put ${\lambda }_j={\pi }_{j}e$. The equality $se=te$ says exactly that $D(u){\lambda }_i={\lambda }_j$ for every morphism $u$, so these morphisms form a categorical cone.

For any other categorical cone $(x_j:X\to D(j))$, the product in a category gives a unique $x:X\to P$ with ${\pi }_{j}x=x_j$. The categorical cone equations give $sx=tx$, so the equalizer gives a unique $\bar{x}:X\to L$ with $e\bar{x}=x$. This is exactly the universal property of a categorical limit. Empty indexing products are terminal objects, so the construction includes the empty diagram in a category. Thus finite products and equalizers give all finite limits.

Choose for each $j\in \mathcal{J}$ a categorical limit $L_j$ of $F(-,j)$, with projections ${\lambda }_{i,j}:L_j\to F(i,j)$. For $v:j\to k$, the morphisms $F(1_i,v){\lambda }_{i,j}$ form a categorical cone over $F(-,k)$, since the two actions of the product functor $F$ commute. Define $L(v):L_j\to L_k$ by the equationsExistence and uniqueness follow from the universal property. The identity morphism of $L_j$ satisfies the equations for $v=1_j$, and $L(w)L(v)$ satisfies those for $wv$, so uniqueness gives $L(1_j)=1_{L_j}$ and $L(wv)=L(w)L(v)$. Hence $L:\mathcal{J}\to \mathcal{D}$ is a functor.

If $L_j^{\prime }$ with projections ${\lambda }_{i,j}^{\prime }$ are other choices, there is a unique isomorphism $u_j:L_j\to L_j^{\prime }$ carrying every projection to the corresponding projection. The same equations show $u_{k}L(v)=L^{\prime }(v)u_j$, because their composites with all ${\lambda }_{i,k}^{\prime }$ agree. Thus the choices give a unique natural isomorphism compatible with the chosen categorical limit categorical cones. Without the compatibility condition, uniqueness of an arbitrary natural isomorphism is not asserted.