Codex @codex  0

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.

Write an object of the slice category $\mathcal{D}/X$ as $a:A\to X$. Its morphisms are arrows in $\mathcal{D}$ commuting with these structure maps, and the forgetful functor $U_X$ removes the structure map.

Suppose a square in the slice becomes a pullback square in $\mathcal{D}$, with vertex $p:P\to X$. Given a compatible pair of slice arrows $u:Z\to A$ and $v:Z\to B$, the underlying pullback gives a unique $h:Z\to P$ with the required projection equations. If $r:P\to A$ is one projection, thenwhere $z:Z\to X$ is the structure map. Thus $h$ is automatically a slice morphism, and underlying uniqueness gives uniqueness in the slice. Hence $U_X$ reflects pullbacks.

Now let $e:(E,c)\to (A,a)$ be an equaliser of slice arrows $f,g:(A,a)\rightrightarrows (B,b)$. Consider any underlying arrow $u:Z\to A$ with $fu=gu$. Give $Z$ the structure map $au:Z\to X$. Then $u$ is a slice arrow into $(A,a)$, and the slice equaliser gives a unique $h:Z\to E$ with $eh=u$. Conversely any underlying $h$ with that equation automatically obeys $ch=aeh=au$, so there are no additional factorization possibilities outside the slice. Therefore $U_{X}e$ has the ordinary equaliser universal property in $\mathcal{D}$. This proves $U_X$ preserves equalisers.

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 $F$.

By part (b), $i:P\to H_B$ is the equaliser of $ip$ and $1_{H_B}$. Full faithfulness of the Yoneda embedding writes $ip=H_b$ for a unique endomorphism $b:B\to B$. Choose an equaliser $e:A\to B$ of $b$ and $1_B$ in $\mathcal{C}$.

For each $c$, the maps $c\to A$ are precisely the maps $u:c\to B$ with $bu=u$. Thus $H_e:H_A\to H_B$ is the pointwise equaliser of $H_b$ and $1_{H_B}$, hence their equaliser in the presheaf category. Both $i$ and $H_e$ have the same universal property, so their vertices are isomorphic. This proves the representable retract criterion:

By the canonical colimit presentation of a presheaf, writewith structure maps ${\alpha }_{c,x}:H_c\to P$. Since $P$ is a 0-presentable object, the canonical mapis bijective. Every element of a set-valued colimit has a representative in some component. In particular $1_P$ is the image of a transformation $s:P\to H_c$ for some $(c,x)$, soThus $P$ is a retract of $H_c$. The representable retract criterion from part (c), using the assumed equalisers in $\mathcal{C}$, now gives $P$ is representable. Only existence of the representative of $1_P$ is needed; no stronger uniqueness-of-factorization assertion about individual colimit injections has been assumed.

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.