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The Category of sets is monoidal under Cartesian product, with any singleton as unit. The tensor of maps is $(f\times g)(x,y)=(f(x),g(y))$, the associator sends $((x,y),z)$ to $(x,(y,z))$, and the left and right unitors discard the singleton coordinate. These are natural bijections. Both pentagon paths send a nested quadruple to the same rebracketed quadruple, and both triangle paths discard the same singleton coordinate, verifying the axioms directly.

This is not the only structure. Disjoint union $X⨿Y$ is another tensor bifunctor, with empty-set unit. Its associator changes parentheses while retaining the tag indicating the original summand, and its unitors remove an empty summand. Every coherence path preserves both the original tag and the element, so again the pentagon and triangle hold.

These product and coproduct monoidal structures on sets are distinct even up to monoidal equivalence. Any equivalence of the underlying category preserves initial and terminal objects up to isomorphism. It therefore cannot carry the empty-set unit of the coproduct structure to the singleton unit of the product structure. Hence Set has at least these two inequivalent monoidal structures.

A terminal object and pullbacks in a category give a binary product $X\times Y$ by pulling back $X\to 1\leftarrow Y$. Repeating this gives every finite product, with the empty product supplied by the terminal object. The preceding parts then construct equalizers of arbitrary parallel arrows.

For a diagram $D:J\to \mathcal{C}$ with finitely many objects and arrows, form $P={\prod }_{j\in \mathrm{Ob}J}D(j)$ and $Q={\prod }_{a:j\to k}D(k)$. Define $r,s:P\to Q$ by $r_a=D(a){\pi }_j$ and $s_a={\pi }_k$. Their equalizer $e:L\to P$ has projections ${\pi }_{j}e$ satisfying every cone equation. Conversely a categorical cone supplies a unique map to $P$, and its cone equations say that this map equalizes $r,s$, hence factors uniquely through $L$. Thus $L$ is the categorical limit. This proves a terminal object and all pullbacks give all finite limits, including the empty diagram.

Assume $F$ preserves finite limits. Given any finite diagram in the comma category $(A\downarrow F)$, first take its underlying limit $c$ in $\mathcal{C}$. Its structure maps $A\to F{c}_j$ form a cone, and preservation gives a unique map $A\to Fc$. This object and the underlying projections satisfy the limiting universal property in the comma category. The construction also includes the empty diagram, since $F$ sends a terminal object to a singleton. Thus $(A\downarrow F)$ has finite limits for every set $A$, proving (ii).

A filtered category is nonempty, any two objects have arrows to a common object, and any parallel pair $u,v:A\rightrightarrows B$ is equalized by some arrow $B\to C$. Equivalently, every finite diagram has a cocone: first choose a common target for its finitely many objects, then successively equalize the finitely many discrepancies along its arrows. A weakly filtered category requires a cocone only for finite nonempty connected diagrams. Equivalently, each connected component of a category is filtered. To see this, two objects in one component are joined by a finite zigzag; a cocone on that zigzag gives a common target, and a parallel-pair diagram gives the equalization condition. Conversely a connected diagram lies in one filtered component. An empty category is weakly filtered, since it has no such diagrams.

The unique outgoing-arrow lifting property is precisely that of a discrete opfibration. Let $F:\mathcal{C}\to \mathcal{D}$ have this property, with $\mathcal{D}$ weakly filtered. Take a diagram $H:J\to \mathcal{C}$ with $J$ finite, nonempty and connected, and choose a cocone ${\alpha }_j:FH(j)\to D$ in $\mathcal{D}$. Lift each ${\alpha }_j$ uniquely from $H(j)$, obtaining ${\tilde{\alpha }}_j:H(j)\to C_j$. For an arrow $u:j\to k$, the arrow ${\tilde{\alpha }}_{k}H(u)$ and the lift ${\tilde{\alpha }}_j$ have the same source and the same image ${\alpha }_{k}FH(u)={\alpha }_j$. Unique lifting gives equality of the arrows and, importantly, $C_j=C_k$. Connectedness of $J$ makes all targets a single object $C_0$. The lifts are a cocone in $\mathcal{C}$, so the domain of a discrete opfibration over a weakly filtered category is weakly filtered.

Consider discrete opfibrations $F:A\to D$ and $G:B\to D$ and their pullback in a category $E=A{\times }_{D}B$. The assumed creation of pullbacks describes its objects as pairs $(a,b)$ with $Fa=Gb$, and its arrows as pairs $(u,v)$ with $Fu=Gv$. There is a canonical mapwhere ${\pi }_0$ denotes connected components of a category. Assume the vertices are weakly filtered categories.

For surjectivity, choose components $[a]$ and $[b]$ whose images belong to the same component of $D$. That component is filtered, so there are arrows $Fa\to d$ and $Gb\to d$. Lifting them from $a$ and $b$ gives $a\to a_1$ and $b\to b_1$ with $F{a}_1=G{b}_1=d$. Thus $(a_1,b_1)\in E$ maps to the chosen component pair.

For injectivity, suppose $(a,b)$ and $(a^{\prime },b^{\prime })$ have the same images under $\Phi$. Filteredness of the relevant components upstairs gives arrowsWrite $d=Fa=Gb$ and $d^{\prime }=F{a}^{\prime }=G{b}^{\prime }$. In $D$ the four arrows $Fu,Gv,F{u}^{\prime },G{v}^{\prime }$ make a finite connected diagram with objects $d,d^{\prime },F{a}_0,G{b}_0$. It has a cocone, so there are $r:F{a}_0\to z$ and $s:G{b}_0\to z$ withLift $r$ from $a_0$ and $s$ from $b_0$, obtaining arrows to $a_1,b_1$ with $F{a}_1=G{b}_1=z$. Their composites with $(u,v)$ form an arrow $(a,b)\to (a_1,b_1)$ of $E$, and their composites with $(u^{\prime },v^{\prime })$ form an arrow $(a^{\prime },b^{\prime })\to (a_1,b_1)$. Hence the two representatives lie in the same component. This proves the connected components of discrete-opfibration pullbacks formulaThe map is the canonical comparison, so this is preservation of the pullback, not just an accidental bijection of its objects.

Now fix a small filtered category $C$. Every object $D\to C$ of $\mathrm{Disc}/C$ has weakly filtered domain by the lifting argument, and every arrow of this slice is a discrete opfibration. Pullbacks in the slice are the created pullbacks above; their resulting domains are weakly filtered as well, since a pullback projection is a discrete opfibration onto a weakly filtered domain. The terminal slice object is $1_C:C\to C$. Because $C$ is nonempty and filtered, it is connected, so ${\pi }_0(C)$ is a singleton. Thus the component functor preserves the terminal object and pullbacks. These construct every finite categorical limit: binary products are pullbacks over the terminal object, and equalizers are pullbacks along a diagonal into a binary product. Therefore the functor from this slice to sets preserves all finite limits.

Take the category of ordinals in reverse order ${\mathbf{Ord}}^{\mathrm{op}}$, with a unique arrow $\alpha \to \beta$ exactly when $\alpha \ge \beta$, and the unique functorBoth categories are complete categories. A small limit in the ordinal category is the supremum of the ordinals occurring in the diagram; the empty supremum is $0$, its terminal object. The functor $G$ preserves all these categorical limits, and both categories are locally small categories.

A left adjoint would select an initial object of ${\mathbf{Ord}}^{\mathrm{op}}$, meaning an ordinal at least as large as every ordinal. No such object exists. More specifically, any set of candidate ordinals has an upper bound $\gamma$, and none of its members maps to $\gamma +1$. Thus the solution-set condition fails. This example isolates the genuine smallness requirement in the general adjoint functor theorem.

Definitions organize category theory, but theorems give that organization mathematical force. I would therefore argue against treating the subject as an exception in which definitions are intrinsically more important than theorems. The distinction is especially misleading here because good categorical definitions are designed to expose hypotheses from which substantial theorems follow.

A category records objects, composable arrows, identities and associative composition. A functor preserves that structure. This language allows one to compare otherwise different mathematical settings, but it does not itself say which constructions exist. For example, the definition of a product in a category is not a recipe for forming a Cartesian product of underlying sets: it requires a universal factorization of pairs of arrows. If two products in a category $(P,p_1,p_2)$ and $(Q,q_1,q_2)$ exist, the universal properties give unique arrows $f:P\to Q$ and $g:Q\to P$ respecting projections. The composites $gf$ and $fg$ respect the same projections as the identities, so uniqueness forces them to be identities. Thus a definition yields a genuine theorem: products in a category are unique up to unique compatible isomorphism. The proof matters because it licenses replacing a construction by any other realization of the same property.

The Yoneda lemma is a more substantial example. For a covariant representable functor $\mathcal{C}(A,-)$ and $X:\mathcal{C}\to \mathbf{Set}$, a natural transformation $t:\mathcal{C}(A,-)\to X$ is determined by $a=t_A(1_A)$. Naturality forcesConversely this formula defines a natural transformation for every $a\in X(A)$. Hence $\mathrm{Nat}(\mathcal{C}(A,-),X)\cong X(A)$. The definition of naturality is indispensable, but the theorem identifies every possible transformation, proves representables detect arrows, and explains why universal objects can be studied through hom-sets. Merely knowing the definition does not supply this conclusion.

Similarly, an adjunction is defined by natural hom-set bijections. The construction of a free abelian group gives $\mathbf{Ab}({\mathbb{Z}}^{(S)},A)\cong \mathbf{Set}(S,UA)$, because a group homomorphism is determined uniquely by its values on the basis. This makes a universal extension problem transparent. But the Freyd general adjoint functor theorem goes further: it turns categorical limit preservation and a solution set into an actual adjoint. Its smallness hypotheses cannot be discarded as bookkeeping. For instance, the underlying-set functor from finite groups to finite sets has no left adjoint at a singleton. A putative universal finite group with universal element of order $m$ could not map that element to a generator of a finite cyclic group of order greater than $m$. The theorem's hypotheses separate existence from attractive notation.

The monoidal coherence theorem offers another example. A monoidal category has associativity and unit isomorphisms satisfying the pentagon identity for a monoidal category and triangle identity for a monoidal category. Coherence proves that every formal reassociation and unit insertion produces the same structural map. This is what justifies calculating without displaying all parentheses; it is not true merely because one has named an associator. Nor does it identify arbitrary additional maps, such as distinct braidings.

Finally, the opposite category construction shows how a definition can economize on proofs: reversing arrows takes products in a category to coproducts in a category, initial objects to terminal objects and monads to comonads. Yet this economy works because theorems and their proofs are invariant under the reversal. The subject is powerful precisely when definitions isolate reusable structure and theorems establish its consequences. Separating their importance obscures that partnership.

Here a pointed complete partial order is a partially ordered set $D$ with a least element ${\perp }_D$ in which every nonempty directed set has a least upper bound. Directed means that every pair of its elements has an upper bound within that set. A Scott continuous map is monotone and preserves these directed suprema. It is not required to preserve the least element; restricting to strict maps would change the categorical claim. If completeness is instead formulated only for increasing countable chains, the same arguments below work with those chains.

Order the function space of complete partial orders $[D\to E]$ pointwise: $f\le g$ means $f(x)\le g(x)$ for every $x$. Its least element is the constant function with value ${\perp }_E$. For a directed family $\mathcal{F}$ of continuous maps, define $h(x)={\sup }_{f\in \mathcal{F}}f(x)$. This is well-defined and is its pointwise least upper bound. It is monotone. For any directed $A\subseteq D$,The interchange is valid because the combined family $\{f(a):f\in \mathcal{F},a\in A\}$ is directed, and either iterated supremum is its least upper bound. Thus $h$ is a Scott continuous map, proving that $[D\to E]$ is a pointed complete partial order.

Finite products carry the product order, componentwise suprema and least element $({\perp }_D,{\perp }_E)$. The one-element order is a terminal object. For the exponential, use the function space with evaluation $\mathrm{ev}(f,x)=f(x)$. To check joint continuity, take a directed family of pairs $(f_i,x_i)$. ThenIndeed, directedness supplies a $k$ above the pairs indexed by both $i$ and $j$, giving $f_i(x_j)\le f_k(x_k)$. Thus the diagonal values are cofinal in the double family, and evaluation preserves its supremum.

For a continuous $h:A\times D\to E$, its currying is $\Lambda h:A\to [D\to E]$, $(\Lambda h)(a)(d)=h(a,d)$. Each section is continuous because $d\mapsto (a,d)$ is; pointwise directed suprema in the $a$ coordinate show that $\Lambda h$ is continuous. Conversely a continuous $g:A\to [D\to E]$ gives the continuous map $\mathrm{ev}\circ (g\times {\mathrm{id}}_D)$. The two constructions are inverse, because they agree on every pair $(a,d)$. This gives the natural bijectionwith the required evaluation map. Together with the terminal object and products, it proves cartesian closure of pointed complete partial orders with non-strict continuous maps.

Every covariant representable functor preserves categorical limits: arrows from its representing object into a limiting object are exactly compatible cones of arrows into the diagram. Apply the density formula proved in the root solution. For any finite diagram $D:J\to \mathcal{C}$,The middle isomorphism uses exactly the permitted theorem that filtered colimits commute with finite limits in sets; it applies because $E^{\mathrm{op}}$ is filtered. All maps are the natural comparison maps, so $F$ preserves finite limits, including the terminal object. This proves $\boxed{\text{(iii)}\Rightarrow \text{(i)}}$ and completes the equivalence.

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.

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 $F$.

A Cartesian closed category has finite categorical products and, for every $P,Q$, an exponential object $Q^P$ with an evaluation morphism $\mathrm{ev}:Q^P\times P\to Q$ such that composition with evaluation gives natural bijectionsFinite categorical products in the presheaf category are pointwise; in particular its terminal object is the constant singleton categorical presheaf. For categorical presheaves $P,Q$ defineFor $u:d\to c$, restriction sends $\alpha$ to $\alpha \circ (H_u\times 1_P)$. Composition and identity laws follow from those of the Yoneda embedding, so this is a categorical presheaf.

Define evaluation at $c$ by ${\mathrm{ev}}_c(\alpha ,x)={\alpha }_c(1_c,x)$. For $u:d\to c$, naturality of $\alpha$ gives $Q(u){\alpha }_c(1_c,x)={\alpha }_d(u,P(u)x)$, which is the evaluation of the restricted pair. Hence evaluation is a natural transformation.

Given $t:R\times P\to Q$, its currying sends $z\in R(c)$ to the transformation whose component at $a$ isNaturality in $a$ follows from that of $t$, and naturality in $c$ follows by composing $v$ with the relevant arrow. Evaluating $\hat{t}$ at $(1_c,x)$ recovers $t_c(z,x)$. Conversely, if $s:R\to Q^P$, naturality of $s$ along $v:a\to c$ says that evaluating $s_a(R(v)z)$ at $(1_a,x)$ gives $(s_c(z){)}_a(v,x)$. Thus currying the evaluation of $s$ recovers $s$. These constructions are inverse and natural, proving the universal property. This is the exponential in a presheaf category, and proves the presheaf category is Cartesian closed.

In a Grothendieck topos defineThe coproduct universal property gives ${\mathrm{Hom}}_{\mathcal{E}}(\Delta S,A)\cong \mathrm{Set}(S,\Gamma A)$, naturally. Thus $\Delta ⊣\Gamma$. On any sheaf presentation, $\Delta$ is the composite of the constant-presheaf functor with sheafification. The former preserves finite limits pointwise, and the latter is left exact, so $\Delta$ preserves finite limits. Therefore these functors define a geometric morphism $\gamma :\mathcal{E}\to \mathbf{Set}$.

For uniqueness, let $v$ be any such geometric morphism. Its inverse image is a left adjoint and preserves the terminal object. Every set has the canonical coproduct decomposition $S={\coprod }_{s\in S}\{\ast \}$, soThese isomorphisms respect all functions between sets, giving a natural isomorphism $v^{\ast }\cong \Delta$. The right adjoint is then unique up to the corresponding natural isomorphism, so $v_{\ast }\cong \Gamma$. There is a unique geometric morphism to sets, up to isomorphism, called the global sections geometric morphism. This also applies to the degenerate topos.