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A poset becomes a category whose objects are its elements and whose hom-set $X(x,y)$ is a singleton if $x\le y$ and empty otherwise. Reflexivity supplies identity arrows and transitivity supplies composition; all possible composites are unique, so the category axioms hold. Antisymmetry says that isomorphic objects are equal. A functor between such categories is exactly an order-preserving function: an inequality must be sent to an inequality, and there is no further choice of arrow map.

The Freyd general adjoint functor theorem is as follows. Let $G:\mathcal{D}\to \mathcal{C}$, with $\mathcal{D}$ a complete category which is locally small. Then $G$ has a left adjoint if and only if it preserves all small categorical limits and satisfies the solution-set condition: for each $c\in \mathcal{C}$, there is a set-indexed familysuch that every $u:c\to Gd$ can be written $u=G(h)u_i$ for some $i$ and some $h:d_i\to d$. The target category need not be complete. We prove all the existence steps.

First establish the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{E}$ be locally small and complete, with a weakly initial set $\{w_i\}$, meaning every object receives a map from at least one $w_i$. Its categorical product $w={\prod }_i{w}_i$ is weakly initial: compose a categorical product projection with the map from the appropriate $w_i$. Local smallness makes $\mathcal{E}(w,w)$ a set. Let $e:v\to w$ be the simultaneous equaliser of every endomorphism of $w$ and the identity. Explicitly it is the equaliser of the two arrowswith components $k$ and $1_w$. Thus $ke=e$ for every endomorphism $k$, and $e$ is monic.

Every object $x$ receives a morphism from $v$, by composing $e$ with a map $w\to x$. To prove uniqueness, take parallel maps $a,b:v\to x$ and their equaliser $m:y\to v$. Weak initiality of $w$ gives $t:w\to y$. The composite $emt$ is an endomorphism of $w$, so $(emt)e=e$. Cancelling the monomorphism $e$ gives $mte=1_v$. Hence $m$ is both monic and a split epimorphism, and is therefore invertible. Since $am=bm$, this forces $a=b$. We have proved $v$ is initial, without using a class-sized categorical product or equaliser.

For sufficiency in the theorem, fix $c$ and consider the comma category $(c\downarrow G)$. Its objects are pairs $(d,u:c\to Gd)$; a morphism $h:(d,u)\to (d^{\prime },u^{\prime })$ satisfies $G(h)u=u^{\prime }$. This category is locally small because each hom-set is a subset of $\mathcal{D}(d,d^{\prime })$. It is complete: for a small diagram of such pairs, take the underlying categorical limit $d$ in $\mathcal{D}$. The arrows from $c$ form a cone to its image under $G$, and preservation of the categorical limit gives a unique $u:c\to Gd$ inducing them. The same universal property gives all mediators in the comma category. The solution-set family is a weakly initial set in this comma category, so the lemma supplies an initial pairFor $f:c\to c^{\prime }$, initiality defines a unique $Ff:Fc\to F{c}^{\prime }$ satisfying $G(Ff){\eta }_c={\eta }_{c^{\prime }}f$. Uniqueness proves the identity and composition laws, making $F$ a functor. Initiality also gives bijectionsThe definition of $Ff$ proves naturality in $c$, and postcomposition proves naturality in $d$. These bijections are the required adjunction $F⊣G$.

For necessity, if $F⊣G$, the single arrow ${\eta }_c:c\to GFc$ is a solution set: the transpose of any $u:c\to Gd$ gives its factorization through ${\eta }_c$. If $d$ is a limiting cone vertex of a small diagram $D_j$ in $\mathcal{D}$, then, naturally in $c$,These are the canonical cone-factorization bijections, so $Gd$ with its image projections is a categorical limit of $G{D}_j$. Thus $G$ preserves all small categorical limits, completing both directions of the theorem.

The general adjoint functor theorem says that, for a functor $U:\mathcal{A}\to \mathcal{B}$ between locally small categories, with $\mathcal{A}$ a complete category, $U$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition. The solution-set condition means that for each $B\in \mathcal{B}$ there is a set of arrows $b_i:B\to U{A}_i$ such that every $b:B\to UA$ factors as $U(a)b_i$ for some $i$ and some $a:A_i\to A$.

Here is the smallness argument behind the theorem. A complete category that is locally small and has a weakly initial set $(W_i)$ has an initial object. Form the product in a category $W={\prod }_i{W}_i$. It is weakly initial, since a projection $W\to W_i$ followed by a chosen arrow $W_i\to X$ gives an arrow $W\to X$. Take the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ with $1_W$; the family is a set by local smallness. The object $E$ is also weakly initial. For parallel arrows $a,b:E\rightrightarrows X$, take their equalizer $j:Y\to E$ and a map $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, the defining property of $e$ gives $ejte=e$. Cancellation of the monomorphism $e$ gives $jte=1_E$. Thus the monomorphism $j$ has a right inverse and is an isomorphism, so $a=b$. Existence of arrows out of $E$ was already established, proving the initial-object lemma for complete categories with a weakly initial set.

If $U$ preserves small categorical limits, the comma category $(B\downarrow U)$ is complete: take the limiting object in $\mathcal{A}$, and use preservation to assemble the arrows from $B$. It is locally small, and the solution-set condition provides a weakly initial set. Its initial object is an arrow ${\eta }_B:B\to ULB$ with a natural bijectionFor a morphism $v:B\to B^{\prime }$, define $Lv$ by $U(Lv){\eta }_B={\eta }_{B^{\prime }}v$. Uniqueness proves functoriality and naturality, giving a left adjoint $L⊣U$. Conversely, a right adjoint preserves categorical limits, because applying the adjunction converts a limiting cone over a diagram into the corresponding limiting cone of hom-sets. Its adjunction unit at $B$ is a singleton solution set. This proves both directions of the general adjoint functor theorem.

Apply it to the inclusion $J:\mathbf{KHaus}\hookrightarrow \mathbf{Top}$, where $\mathbf{KHaus}$ is the full subcategory of compact Hausdorff spaces. A small product in a category of compact Hausdorff spaces is again compact Hausdorff by the Tychonoff theorem. An equalizer of two continuous functions into a Hausdorff space is closed, hence compact Hausdorff. Products and equalizers construct all small categorical limits, and the inclusion preserves these constructions. Both categories are locally small.

Fix a topological space $X$ and a continuous function $f:X\to K$ into a compact Hausdorff space. The closure $K_f=\overline{f(X)}$ is compact Hausdorff and has a dense subset of cardinality at most $|X|$. The permitted cardinal estimate givesFor $X=\varnothing$, use $K_f=\varnothing$. Choose a set of representatives of all compact Hausdorff topologies on underlying sets of cardinality at most this bound, and all continuous functions from $X$ to those representatives. Every $f$ factors through one of them, using the inclusion $K_f\hookrightarrow K$. This is the solution-set condition. Consequently the inclusion of compact Hausdorff spaces has a left adjoint, the compact Hausdorff reflection:The universal arrow $X\to RX$ has dense image: factoring it through its closed image and applying uniqueness supplies an inverse to that image inclusion. It need not be an embedding when $X$ is not sufficiently separated; the assertion is a compact Hausdorff reflection, rather than a claim that every topological space embeds in a compact Hausdorff space.

For completeness, the alternative Special adjoint functor theorem can be proved from the same initial-object argument. Its limit form assumes that $\mathcal{A}$ is complete, locally small and well-powered, with a small cogenerating family $(Q_i{)}_{i\in I}$; then $U:\mathcal{A}\to \mathcal{B}$, with $\mathcal{B}$ locally small, has a left adjoint precisely when it preserves small limits. To prove sufficiency, start with $b:B\to UA$ and intersect all subobjects $M\hookrightarrow A$ through which $b$ factors after applying $U$. Well-poweredness makes this a small intersection of subobjects, and preservation of limits gives a smallest supporting object $(M,b_M)$. For maps $s,t:M\rightrightarrows Q_i$, equality $U(s)b_M=U(t)b_M$ makes their equalizer another supporting subobject. Minimality forces that equalizer to be invertible, so $s=t$. Thereforeis an injective function. The evaluation embedding into cogenerator products embeds $M$ in ${\prod }_i{Q}_i^{S_i}$, where $S_i$ is the realized subset of the fixed set $\mathcal{B}(B,U{Q}_i)$. There are only a set of choices of these subsets, only a set of subobjects of each resulting product, and only a set of arrows from $B$ to each $U$-image. They give a solution set, so the already proved general adjoint functor theorem applies. Necessity is again preservation of limits by a right adjoint. For $\mathbf{KHaus}$, monomorphisms are embeddings onto closed subspaces, and $[0,1]$ is a coseparator because continuous functions to it separate points. Thus the Special adjoint functor theorem gives the same reflection directly.

The Yoneda embedding is an exact representation of a category, but does not by itself reduce the subject to the study of arbitrary functor categories. There are both a size issue and a structure issue.

Let $\mathcal{A}$ be a locally small category and set $yA=\mathcal{A}(-,A)$. For a categorical presheaf $P:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, a natural transformation $\theta :yA\to P$ is determined by $x={\theta }_A(1_A)$. Indeed, naturality along $f:B\to A$ forcesConversely this formula defines a natural transformation for each $x\in P(A)$, since functoriality gives naturality along every $g:C\to B$. Taking $P=y{A}^{\prime }$ givesThus the Yoneda embedding is full and faithful, and $\mathcal{A}$ is equivalent to its full subcategory of representable presheaves. The indexing category in the usual version is ${\mathcal{A}}^{\mathrm{op}}$, not necessarily a small category chosen independently of $\mathcal{A}$.

Local smallness makes the values $\mathcal{A}(B,A)$ sets; it does not make the collection of objects a set. For a large indexing category, the collection of natural transformations between arbitrary set-valued functors can itself be a proper class. For example, take the discrete category on the class of ordinals and the constant two-element functor. Its endomorphisms include an independently chosen function $\{0,1\}\to \{0,1\}$ at each ordinal. A small functor category cannot encode this collection without changing universes. The Yoneda lemma remains meaningful when the source is representable because the displayed bijection identifies its natural transformations with a set. It does not make the entire large functor category locally small.

Even for a small category, the representable image is usually a very special part of its presheaf category. If $\mathcal{A}$ is the terminal category, its presheaf category is $\mathbf{Set}$, but its representable image consists only of a singleton. The presheaf category has all small categorical limits and colimits, whereas an arbitrary full subcategory need have almost none. The Yoneda embedding preserves existing limits because representable functors preserve limits. It generally does not preserve colimits. For example, the coproduct of two singleton sets in category of finite sets is a two-element set, but evaluating the proposed presheaf coproduct at a two-element set givesThe pointwise coproduct therefore leaves the representable image. Identifying which presheaves are representable, and which ambient constructions return to the original category, is precisely substantive category theory.

The representation is nonetheless powerful. The Yoneda lemma replaces a morphism by all its probes, converts a universal property into a representation of a functor, and makes the presheaf category a natural place to add formal colimits. A small presheaf category is the free free cocompletion under small colimits: every presheaf is a colimit of representables indexed by its category of elements. But this is a controlled enlargement of the category, not an identification of its original objects with all functors. The stronger conclusion in the proposed assertion therefore does not follow from the full-faithfulness statement.

A balanced category is one in which every morphism that is both a monomorphism and an epimorphism is an isomorphism. A strong monomorphism $m:A\to B$ is a monomorphism right orthogonal to all epimorphisms: whenever $e:X\to Y$ is epic and $a:X\to A$, $b:Y\to B$ satisfy $ma=be$, there is a unique $d:Y\to A$ with $de=a$ and $md=b$. A regular monomorphism is an equalizer of some parallel pair of morphisms. Its equalizer property supplies the indicated lift after epic cancellation, so every regular monomorphism is strong.

If every monomorphism is strong and $m:A\to B$ is also epic, apply strongness to the square whose left and right sides are both $m$, with horizontal arrows $1_A$ and $1_B$. The lift $d:B\to A$ satisfies $dm=1_A$ and $md=1_B$. Thus $m$ is invertible and the category is balanced.

Conversely, suppose the category is balanced and has pullbacks in a category. Given the lifting square above, form $P=A{\times }_{B}Y$ with projections $p:P\to Y$, $q:P\to A$. The pair $(a,e)$ gives $h:X\to P$ with $ph=e$, $qh=a$. The pullback of the monomorphism $m$ is monic, so $p$ is monic. It is also epic: $rp=sp$ implies $re=se$, hence $r=s$ because $e$ is epic. Balancedness makes $p$ invertible. Now $d=q{p}^{-1}$ satisfies both required equations, and monicity of $m$ proves uniqueness. Hence in a balanced category with pullbacks, every monomorphism is strong.

For the finite example, choose a strong monomorphism $m:A\to B$ that is not regular. It is not an isomorphism, since an isomorphism is the equalizer of an identical pair. It is not epic either, since an epic strong monomorphism is invertible by the first argument. Consequently there are distinct arrows $r,s:B\rightrightarrows D$ with $rm=sm$. Since $m$ is not the equalizer of this pair, there is a map $h:X\to B$ with $rh=sh$ that does not factor through $m$. Monicity rules out failure of uniqueness, so failure of existence is the only possible failure of its equalizer property.

Define ${\mathcal{C}}_0$ to have four formally distinct objects $a,b,x,d$ and exactly the following six nonidentity arrows:Their only nontrivial composites areThere are no longer composable strings of nonidentity arrows, so these rules define an associative category. Sending $a,b,x,d$ to $A,B,X,D$, respectively, and the six arrows to $m,h,r,s,rm,rh$, defines a functor. It is a faithful functor: every hom-set has at most one arrow except ${\mathcal{C}}_0(b,d)=\{r_0,s_0\}$, and their images are distinct. Faithfulness does not require the four image objects to be distinct.

The map $m_0$ is monic, by direct inspection of arrows into $a$. Neither $m_0$ nor $h_0$ is epic, since the distinct pair $r_0,s_0$ agrees after either. Every other nonidentity arrow has target $d$ and is epic, because there is only one arrow out of $d$. A lifting square against $m_0$ whose left side is one of these epimorphisms would require an arrow $d\to b$, which does not exist. Squares whose left side is an identity have the evident unique lift. Thus $m_0$ is a strong monomorphism.

It is not a regular monomorphism. The only distinct parallel pair out of $b$ is $r_0,s_0$, and $h_0$ equalizes it but has no factorization through $m_0$, since there is no arrow $x\to a$. An identical pair has $1_b$ as its equalizer and cannot have the noninvertible $m_0$ as its equalizer. Therefore

Use the elementary topos definition: a category with finite limits, exponential objects, and a subobject classifier. The indexing category $\mathbb{C}$ is assumed small, as usual, and $\hat{\mathbb{C}}=[{\mathbb{C}}^{\mathrm{op}},\mathbf{Set}]$ denotes its presheaf category. A finite limit of categorical presheaves is formed objectwise: at $c$ take the corresponding limit of sets, and apply the universal property to induce restriction maps. The terminal categorical presheaf is the constant singleton, so this supplies all required finite limits.

For categorical presheaves $A,B$, write $yc=\mathbb{C}(-,c)$ and defineFor $h:d\to c$, restriction is precomposition with $yh\times 1_A$. Evaluation sends $(\theta ,a)$ at $c$ to ${\theta }_c(1_c,a)$. Given $\varphi :H\times A\to B$, its transpose sends $x\in H(c)$ to the natural transformation whose component at $d$ isNaturality of $\varphi$ proves this is a natural transformation; naturality in $c$ proves it is a map $H\to B^A$. Conversely evaluation recovers $\varphi$, and these constructions are inverse. Thereforeso the presheaf category is Cartesian closed.

Define $\Omega (c)$ to be the set of sieves on a category with codomain $c$. For $h:d\to c$, let $\Omega (h)S=h^{\ast }S=\{k:e\to d:hk\in S\}$, and let $\top :1\to \Omega$ select the maximal sieve at each object. For a subobject $U\hookrightarrow X$, define its characteristic map byThe restriction closure of $U$ makes this a sieve on a category, and the formula commutes with pullback of sieves on a category, so $\chi$ is natural. It is maximal exactly when $1_c$ belongs to it, equivalently when $x\in U(c)$; hence the pullback of $\top$ along $\chi$ is $U$. Uniqueness is also explicit: naturality forces $h\in {\chi }_c(x)$ exactly when the restriction $X(h)x$ lies in the truth pullback at $d$. Thus the sieve presheaf is a subobject classifier, and every small presheaf category is an elementary topos.

Let $D:J\to \mathcal{C}$ be a finite diagram in a category. Form the finite products in a categoryDefine $s,t:P\rightrightarrows Q$ by ${\pi }_{a}s=D(a){\pi }_j$ and ${\pi }_{a}t={\pi }_k$. Take their equalizer $e:L\to P$. The maps ${\lambda }_j={\pi }_{j}e$ satisfy $D(a){\lambda }_j={\lambda }_k$, so they constitute a categorical cone.

Any other categorical cone $x_j:X\to D(j)$ gives a unique $x:X\to P$ by the product in a category property. Its cone equations say $sx=tx$, so the equalizer gives a unique $\bar{x}:X\to L$ with $e\bar{x}=x$. The projections then give ${\lambda }_j\bar{x}=x_j$, and uniqueness follows from both universal properties. Thus this equalizer of two product maps is the finite limit of the diagram in a category.

The empty indexing category is included: its empty product in a category is a terminal object, which is the empty categorical limit. Hence no nonemptiness restriction has entered the construction.

An object of the slice category $\mathcal{C}/A$ is a morphism $x:X\to A$; a map from $x$ to $y:Y\to A$ is $u:X\to Y$ with $yu=x$. The identity morphism $1_A:A\to A$ is a terminal object of this slice category, even when $\mathcal{C}$ itself has no terminal object.

For a cospan $X\stackrel{u}{\to }Z\stackrel{v}{\leftarrow }Y$ in the slice category, form its underlying pullback in a category $P$ in $\mathcal{C}$, with maps $p:P\to X$ and $q:P\to Y$. If $x,y,z$ are the structure maps to $A$, then $xp=zup=zvq=yq$. This common map equips $P$ with its slice structure. An underlying mediating map into $P$ is automatically over $A$, because its composite with $xp$ is the prescribed structure map. Hence the same square is a pullback in a category in $\mathcal{C}/A$.

Part (b) applies to the slice's terminal object and pullbacks in a category. Every slice category has all finite limits when the ambient category has pullbacks in a category. The finite completeness of slice categories result needs no ambient terminal-object hypothesis.

If every monomorphism is strong, let $m:A\to B$ be both a monomorphism and an epimorphism. Use $m$ as both vertical maps in the lifting square, with horizontal maps $1_A$ and $1_B$. The strong monomorphism property gives $d:B\to A$ satisfying $dm=1_A$ and $md=1_B$. Thus $m$ is an isomorphism, proving that the category is balanced.

For the converse, take $e:X\to Y$ epic, $m:A\to B$ monic, and $mu=ve$. Form the pullback in a category of $m$ along $v$:The equality $mu=ve$ gives $t:X\to P$ with $pt=e$ and $qt=u$. By preservation of monomorphisms under pullback in a category, $p$ is monic. It is also epic: if $hp=kp$, then $he=hpt=kpt=ke$, and cancellation of $e$ gives $h=k$. The balanced category hypothesis therefore makes $p$ an isomorphism.

Set $d=q{p}^{-1}$. Then $md=v$ and $de=qt=u$. Any other such lift agrees with $d$ after composing with the monomorphism $m$, so it is equal to $d$. Consequently in a category with pullbacks in a category, being balanced is equivalent to every monomorphism being a strong monomorphism. This is the balanced categories with pullbacks have strong monomorphisms principle; the proof does not assume pullback stability of epimorphisms.

The limit form of the Special adjoint functor theorem is as follows. Let $\mathcal{C}$ be a locally small category, a complete category, and a well-powered category, and suppose it has a small cogenerating family $(Q_i{)}_{i\in I}$. Let $\mathcal{D}$ be a locally small category. Then a functor $G:\mathcal{C}\to \mathcal{D}$ has a left adjoint if and only if it preserves all small categorical limits. The dual exchanges completeness, well-poweredness and cogenerators for cocompleteness, well-copoweredness and generators, and characterizes functors with a right adjoint.

Here a cogenerating set distinguishes unequal parallel maps by postcomposition into one of the $Q_i$. A well-powered category has a set of subobjects of each fixed object. Completeness means existence of all small categorical limits. We use the initial-object lemma for complete categories with a weakly initial set: a complete category that is locally small has an initial object exactly when it has a weakly initial set. A proof is given in Question 5(c). We also use the fact that when $G$ preserves small categorical limits, the comma category $(B\downarrow G)$ is complete, with its limits constructed in $\mathcal{C}$; local smallness is inherited from $\mathcal{C}$. Finally, having an initial object in $(B\downarrow G)$ for each $B$ is the universal arrow from an object to a functor characterization of a left adjoint.

Assume $G$ preserves small categorical limits, and fix $B\in \mathcal{D}$. We will construct a weakly initial set in $(B\downarrow G)$, rather than assume a solution set. Start with an arbitrary object $(C,x)$, where $x:B\to GC$. Call a subobject $m:S\hookrightarrow C$ supporting if $x=G(m)x_S$ for some $x_S:B\to GS$. There is at least one, namely $1_C$. Because $G$ preserves pullbacks in a category, it preserves monomorphisms: the self-pullback characterization of a monic map is preserved. Consequently each $x_S$ is unique.

There is a set of supporting subobjects by well-poweredness. Their intersection of subobjects exists by completeness; write it $m_0:C_0\hookrightarrow C$. Applying $G$ to this intersection categorical limit gives a unique $x_0:B\to G{C}_0$ lifting all the $x_S$, and hence $x=G(m_0)x_0$. Minimality says that any subobject $n:R\hookrightarrow C_0$ supporting $x_0$ is invertible: $m_{0}n$ supports $x$, so $m_0$ factors through $m_{0}n$, giving a right inverse to the monic $n$.

If $u,v:C_0\to Q_i$ satisfy $G(u)x_0=G(v)x_0$, their equalizer supports $x_0$, since $G$ preserves that equalizer. It is therefore an isomorphism, and $u=v$. We have obtained an injective functionLet $J_i$ be its image, a subset of the fixed set $S_i=\mathcal{D}(B,G{Q}_i)$. The evaluation embedding into cogenerator products is a monomorphismIndeed each $s\in J_i$ corresponds to a unique map $C_0\to Q_i$, and the cogenerating set property makes the resulting family jointly monic. Notice that we use the realized subsets $J_i$, not all of $S_i$: there need not be a map $C_0\to Q_i$ available for an unused index.

There is a set of tuples $J=(J_i)$, since $I$ and every $S_i$ are sets. For each resulting $P_J$, choose representatives of its subobjects; these form a set by well-poweredness. For every representative $R$, include every map $y:B\to GR$, a set by local smallness of $\mathcal{D}$. All resulting pairs $(R,y)$ form a set. Our given $(C,x)$ receives a map from such a pair: transport $x_0$ along the isomorphism between $C_0$ and the chosen representative, then compose its inverse with $m_0$. Thus these pairs form a weakly initial set in $(B\downarrow G)$. This is the cogenerator bound for comma-category solution sets.

The initial-object lemma for complete categories with a weakly initial set gives an initial object $(LB,{\eta }_B)$. For $h:B\to B^{\prime }$, define $Lh$ by the unique equationUniqueness proves the functor laws and gives the natural bijectionsThus $L⊣G$. Conversely, a right adjoint preserves small categorical limits: its adjunction identifies maps into the proposed limiting object with compatible families of maps into the diagram. This proves both directions of the limit form of the special adjoint functor theorem. Applying the argument to the opposite categories proves the dual statement. Choices of representatives and adjoint objects are understood in the usual ambient-universe convention for large categories.

A representation of a functor $(A,x)$ for $G$ says precisely that for every object $(B,y)$ of $(1\downarrow G)$ there is a unique arrow $f:A\to B$ with $Gf(x)=y$. This is exactly the initial object property for $(A,x)$ in that comma category. Conversely, an initial object supplies these unique arrows, hence the representing bijections $\mathcal{C}(A,B)\cong GB$.

If $G$ is representable, it is a limit-preserving functor, and its universal element gives an initial object of $(1\downarrow G)$, hence a weakly initial singleton.

Conversely, a limit-preserving $G$ makes $(1\downarrow G)$ a complete category. For a small diagram $(A_j,x_j)$, take $L={\lim }_j{A}_j$ in $\mathcal{C}$. Its distinguished elements form a compatible family in ${\lim }_{j}G{A}_j$. Categorical limit preservation gives a unique $x\in GL$ with $G{p}_j(x)=x_j$. The underlying categorical limit factorization of a categorical cone preserves this element, proving the comma-category universal property. For the empty diagram, this uses $G1\cong 1$. This is the construction of limits in a comma category of a limit-preserving functor.

The comma category is locally small since its arrows are subsets of the hom-sets in $\mathcal{C}$. By the previous part, its weakly initial set therefore yields an initial object. Part (b) then yields a representation of $G$. Thus representability from a solution set follows with all small categorical limits, rather than finite categorical limits alone.

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.

An isomorphism $f:A\to B$ is a morphism with an inverse $f^{-1}:B\to A$ satisfying $f^{-1}f=1_A$ and $f{f}^{-1}=1_B$. A groupoid is a category in which every morphism is an isomorphism.

With the given one-sided inverses,Consequently $f$ is an isomorphism with $f^{-1}=g=h$. This argument uses only the category axioms.

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 $L$.

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.