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Define the forgetful functor $G^{\mathbb{T}}(A,a)=A$, acting as the identity on underlying morphisms. Define the free algebra functor byThe monad laws make ${\mu }_X$ an algebra for a monad structure, and naturality of $\mu$ makes $T(h)$ an monad algebra morphism.

For $(A,a)$, define the transpose mapsThe map $\Psi (k)$ is an monad algebra morphism, because naturality of $\mu$ and the algebra associativity law giveFor $k:X\to A$, naturality of $\eta$ gives $\Phi \Psi (k)=aT(k){\eta }_X=a{\eta }_{A}k=k$. Conversely, if $h$ is an monad algebra morphism, thenPrecomposing by a map into $X$ or postcomposing by an monad algebra morphism respects both formulas. Thus the bijection is natural and $F^{\mathbb{T}}⊣G^{\mathbb{T}}$. This is the free-forgetful Eilenberg-Moore adjunction.

Its adjunction unit is the given ${\eta }_X$; its adjunction counit at $(A,a)$ is the monad algebra morphism $a:(TA,{\mu }_A)\to (A,a)$. Hence $G^{\mathbb{T}}{F}^{\mathbb{T}}=T$, and the induced multiplication is the underlying counit at $(TX,{\mu }_X)$, namely ${\mu }_X$. Therefore this adjunction induces exactly the original monad, including its unit and multiplication of a monad.

Take any morphism $h:A\to B$ and let $c:B\to C$ be its categorical cokernel. Let $i:I\to B$ be the categorical kernel of $c$. Since $ch=0$, kernel universality gives $e:A\to I$ with $h=ie$. The map $i$ is a monomorphism. We must establish the pseudo-epimorphism property of $e$, rather than merely name $I$ an image.

Suppose $t:I\to Z$ satisfies $te=0$. Let $k:K\to I$ be its categorical kernel. Then $e=ka$ for some $a:A\to K$. The composite $ik$ is a monomorphism, so the hypothesis that every monomorphism is normal provides a map $q:B\to D$ of which $ik$ is a categorical kernel, up to its unique compatible isomorphism. Now $qh=qika=0$. The categorical cokernel property of $c$ gives $q=\bar{q}c$, and therefore $qi=\bar{q}ci=0$.

Since $ik$ is the categorical kernel of $q$, the map $i$ factors through it: $i=ik\ell$ for some $\ell :I\to K$. Cancel the monomorphism $i$ to get $k\ell =1_I$. As $k$ is also a monomorphism, $\ell k=1_K$, so $k$ is an isomorphism. Finally $t=tk\ell =0$, proving that $e$ is a pseudo-epimorphism.

ThusThe pseudo-epimorphism factorization through the kernel of a cokernel proof uses a pointed category, existence of categorical kernels and categorical cokernels, and normality of every monic map. It does not silently replace pseudo-epic cancellation against zero by the stronger epimorphism condition.

The comma category $(X\downarrow G)$ has objects $(A,x)$ with $A\in \mathcal{C}$ and a function $x:X\to GA$. A morphism $(A,x)\to (B,y)$ is an arrow $f:A\to B$ such that $Gfx=y$. Identities and composition come from $\mathcal{C}$. When $X=1$, a map $1\to GA$ selects an element of $GA$, so this is the covariant category of elements of $G$.

For the monad $(T,\eta ,\mu )$, an algebra for a monad is $(A,a:TA\to A)$ with $a{\eta }_A=1_A$ and $aT(a)=a{\mu }_A$. A morphism of algebras for a monad $f:(A,a)\to (B,b)$ satisfies $fa=bTf$. These objects and arrows form the Eilenberg-Moore category ${\mathcal{C}}^T$.

The free algebra functor sends $A$ to $(TA,{\mu }_A)$ and $f$ to $Tf$. The monad identities verify the algebra laws. For the forgetful functor $U$, the free adjunction iswith inverse $v\mapsto bTv$. The algebra law makes the latter an monad algebra morphism. The identities $h{\eta }_A\mapsto bT(h{\eta }_A)=h{\mu }_{A}T{\eta }_A=h$ and $bTv{\eta }_A=b{\eta }_{B}v=v$ prove the bijection.

In the category of adjunctions inducing a fixed monad, objects are adjunctions $F⊣G$ with their induced monad identified with $T$. A morphism to $F^{\prime }⊣G^{\prime }$ is a functor $H$ between the right-hand categories satisfying $G^{\prime }H=G$, $HF=F^{\prime }$ and compatibility with units and counits. With these strict identifications, defineThe triangle identities give the unit algebra law; naturality of $\epsilon$ at ${\epsilon }_B$ gives the multiplication law. Naturality at $f$ makes $Gf$ an monad algebra morphism. Moreover $UK=G$, $KF=F^T$ because $G{\epsilon }_{FA}={\mu }_A$, and $K{\epsilon }_B$ is the free-adjunction counit at $K(B)$. Hence $K$ is a morphism into the Eilenberg-Moore adjunction.

For any other such $H$, its underlying object at $B$ must be $GB$. Counit compatibility forces its algebra action to be $G{\epsilon }_B$, and the forgetful functor, which is a faithful functor, forces $H(f)=Gf$. Thus $H=K$. This proves terminality of the Eilenberg-Moore adjunction. If adjunctions are specified only up to coherent isomorphisms, the same argument gives uniqueness up to the corresponding compatible natural isomorphism.

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 $\mathcal{B}$.

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 $J\times K$.

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.