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Let $D:I\to \mathcal{C}$ be a finite diagram. Form the finite categorical productsThere are two morphisms $s,t:P\rightrightarrows Q$: their components at $u:i\to j$ are $D(u){\pi }_i$ and ${\pi }_j$, respectively. Let $e:L\to P$ be their equaliser. Then ${\ell }_i={\pi }_{i}e$ satisfy $D(u){\ell }_i={\ell }_j$, so they constitute a cone over a diagram.

Any other cone $x_i:X\to D(i)$ uniquely determines $x:X\to P$. Its cone equations say precisely that $sx=tx$. The equaliser therefore gives a unique $h:X\to L$ with $eh=x$, and thus ${\ell }_{i}h=x_i$. Conversely these component equations force $eh=x$ by the categorical product universal property, so there is exactly one such mediator. This proves that $(L,{\ell }_i)$ is a categorical limit of $D$.

The empty categorical products are included: if $I$ is empty, $P$ and $Q$ are terminal and the construction gives the empty categorical limit. Thus finite categorical products and equalisers give every finite categorical limit, as a special case of the construction of small limits from products and equalizers.

Let $1$ be a terminal object. The pullback of the unique arrows $A\to 1\leftarrow B$ is a binary categorical product: maps $X\to A$ and $X\to B$ automatically have equal composites into $1$, so the pullback universal property is exactly the categorical product universal property.

For parallel arrows $f,g:A\rightrightarrows B$, form $B\times B$ and take the pullback of $⟨f,g⟩:A\to B\times B$ along the diagonal $\delta :B\to B\times B$. Write the resulting projections as $e:E\to A$ and $b:E\to B$. The pullback equation givesIf $u:X\to A$ satisfies $fu=gu$, the pair $(u,fu)$ is a pullback cone, producing a unique $v:X\to E$ with $ev=u$ and $bv=fu$. The second condition is forced by the first, since $b=fe$. Thus $e$ is the equaliser of $f,g$.

The terminal object and iterated binary categorical products give all finite categorical products, including the nullary one. Part (a) therefore proves pullbacks and a terminal object give all finite categorical limits.

Take any limiting cone ${\ell }_i:L\to D(i)$ in $\mathcal{C}$ of the given shape. Since $GF$ preserves such categorical limits, the cone $GF{\ell }_i:GFL\to GFD(i)$ is limiting in $\mathcal{E}$. It is the image under $G$ of the cone $F{\ell }_i$ in $\mathcal{D}$. Reflection by $G$ therefore says that $F{\ell }_i$ is limiting in $\mathcal{D}$. Since this works for every diagram and every limiting cone of the specified shape, $F$ preserves categorical limits of that shape.

Let ${!}_A:A\to 1$ be the unique arrow to the terminal object of $\mathcal{C}$. DefineThis is a functor into the slice category $\mathcal{D}/F1$, because ${!}_{B}f={!}_A$, and $U_{F1}\hat{F}=F$. The terminal object of this slice is $1_{F1}:F1\to F1$, exactly $\hat{F}(1)$, so $\hat{F}$ preserves the terminal object.

The composite $U_{F1}\hat{F}$ preserves pullbacks by hypothesis, and $U_{F1}$ reflects pullbacks by part (b). Part (a), restricted to the pullback shape, therefore shows that $\hat{F}$ preserves pullbacks. The allowed finite-limit criterion now gives that $\hat{F}$ preserves all finite categorical limits, and hence equalisers. Since $U_{F1}$ preserves equalisers, their composite does too. ThusThis is equalizer preservation by pullback-preserving functors. The factorization supplies terminal-object preservation only in the slice; none is asserted for the original $F$.

For a diagram shape $I$, a functor $F:\mathcal{C}\to \mathcal{D}$ preserves categorical limits of shape $I$ if every limiting cone $(L,{\ell }_i)$ over every diagram $D:I\to \mathcal{C}$ is carried to a limiting cone $(FL,F{\ell }_i)$ over $FD$. It reflects categorical limits of that shape if a cone in $\mathcal{C}$ is limiting whenever its image cone is limiting. These properties concern cones which are given; reflection is not an assertion that every cone in the target lifts.

Suppose $F$ is a full and faithful functor and the image of $(L,{\ell }_i)$ is limiting. A cone $(X,x_i)$ in $\mathcal{C}$ becomes a cone in $\mathcal{D}$, so there is a unique $h:FX\to FL$ with $(F{\ell }_i)h=F{x}_i$. Fullness writes $h=Fk$ for some $k:X\to L$. Faithfulness gives ${\ell }_{i}k=x_i$, and also ensures uniqueness: two such $k$ would have the same image, by uniqueness of $h$. Hence the original cone is limiting. This proves every full and faithful functor reflects categorical limits, the limits reflected by full and faithful functors criterion, with no essential-surjectivity requirement.

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.

Write $\hat{\mathcal{C}}=[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ and similarly for $\mathcal{D}$. The induced inverse-image functor is precomposition:All categorical limits in a presheaf category are computed pointwise. Thus $u^{\ast }$ preserves every limit, in particular finite limits.

Because the indexing categories are small and $\mathbf{Set}$ is complete and cocomplete, both Kan extensions along $f^{\mathrm{op}}$ exist. Their universal properties giveFor example their values are the comma-category formulasThe comma categories here are formed in ${\mathcal{D}}^{\mathrm{op}}$: an object of the first involves $f(c)\to d$ there, equivalently $d\to f(c)$ in $\mathcal{D}$. An object of the second involves $d\to f(c)$ there, equivalently $f(c)\to d$ in $\mathcal{D}$. Keeping the opposite category explicit prevents reversal of the two constructions.

The right-hand adjunction and finite-limit preservation are exactly the axioms for a geometric morphism $u:\hat{\mathcal{C}}\to \hat{\mathcal{D}}$. The extra left adjoint makes it an essential geometric morphism. On representable presheaves, its left adjoint satisfies $u_{!}(y_{\mathcal{C}}c)\cong y_{\mathcal{D}}(fc)$, by the Yoneda lemma and the left adjunction. This identifies the induced morphism directly with the original functor $f$.

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.