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For a locally small category $\mathcal{C}$, an object $A$, and a functor $F:\mathcal{C}\to \mathbf{Set}$, the covariant form of the Yoneda lemma is the natural bijectionFor $x\in F(A)$ define ${\alpha }_B^x(f)=F(f)(x)$, where $f:A\to B$. For $u:B\to D$, functoriality gives $F(u){\alpha }_B^x(f)=F(uf)(x)={\alpha }_D^x(uf)$, so this is a natural transformation. Conversely, naturality of $\alpha$ at $f:A\to B$ gives ${\alpha }_B(f)=F(f){\alpha }_A(1_A)$. The two constructions are inverse. They are natural in $F$, because a natural transformation $F\Rightarrow G$ commutes with evaluation. They are also natural in $A$: an arrow $v:A^{\prime }\to A$ induces $\mathcal{C}(A,-)\to \mathcal{C}(A^{\prime },-)$ by precomposition, and the resulting map $\mathrm{Nat}(\mathcal{C}(A^{\prime },-),F)\to \mathrm{Nat}(\mathcal{C}(A,-),F)$ corresponds to $F(v):F(A^{\prime })\to F(A)$. Applying the result to the opposite category gives $\mathrm{Nat}(\mathcal{C}(-,A),H)\cong H(A)$ for $H:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$.

Now let $\mathcal{C}$ be a small category, and let $F,G:\mathcal{C}\to \mathbf{Set}$. Finite products in a category in $[\mathcal{C},\mathbf{Set}]$ exist pointwise; the terminal functor is the constant singleton functor. Define the exponential of covariant set-valued functors bySmallness ensures that these natural transformations form a set. For $u:C\to D$, define $(G^F)(u)(\alpha )$ at $E$ byPrecomposition respects identities and composition, so this defines a functor. Its evaluation map of an exponential object isTo check naturality, for $u:C\to D$ both sides give ${\alpha }_D(u,F(u)x)$: one side uses naturality of $\alpha$, and the other uses the defining action of $G^F$.

Here is the full universal property. Given $\theta :H\times F\Rightarrow G$, define $\hat{\theta }:H\Rightarrow G^F$ byFor $w:D\to E$, naturality of $\theta$ makes this commute with $G(w)$, so ${\hat{\theta }}_C(h)$ is a natural transformation. Replacing $h$ by $H(u)h$ is the same as precomposing $v$ by $u$, proving naturality in $C$. Conversely, $\beta :H\Rightarrow G^F$ gives ${\theta }_C(h,x)={\beta }_C(h{)}_C(1_C,x)$. Starting with $\theta$ recovers it immediately. Starting with $\beta$, its naturality at $v:C\to D$ yieldsso it too is recovered. The formulas are natural in $H$ and $G$, giving the required adjunction $(-)\times F⊣(-{)}^F$. Therefore the covariant functor category is cartesian closed. This proof supplies the actual exponential and does not need a colimit presentation of its arguments.

Finally suppose $\mathcal{C}$ itself is a small category which is a Cartesian closed category. In its presheaf category, the same construction, applied to ${\mathcal{C}}^{\mathrm{op}}$, givesThe first isomorphism uses the product's universal property: a map into $C\times A$ is precisely a pair of maps into $C$ and $A$. The next is the Yoneda lemma, and the last is the exponential adjunction in $\mathcal{C}$. Each step is natural in $C$, $A$, and $B$. Under these identifications, evaluation corresponds to $Y$ of the evaluation $B^A\times A\to B$. Hence the Yoneda embedding preserves exponentials, with the natural identification $\boxed{Y(B^A)\cong (YB{)}^{YA}}$.

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.

We use the general adjoint functor theorem in its solution-set form. Let $G:\mathcal{D}\to \mathcal{C}$ be a functor between locally small categories, with $\mathcal{D}$ a complete category. Then $G$ has a left adjoint exactly when it preserves all small categorical limits and, for each $C\in \mathcal{C}$, there is a set-indexed familysuch that every $u:C\to GD$ factors as $Gf{u}_i$ for some $i$ and some $f:D_i\to D$. This is the solution-set condition. The indexing family is a set, whereas the entire comma category $(C\downarrow G)$ can be large.

We first prove 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 $(E_i{)}_{i\in I}$. Its product in a category $W={\prod }_i{E}_i$ is weakly initial: follow a projection by a map from the corresponding $E_i$ to any desired object. The hom-set $\mathcal{E}(W,W)$ is a set, so we can form the simultaneous equalizerThis is a small limit; equivalently, equalize the family of endomorphisms with the family of identity maps into a product of copies of $W$. Every object receives a map from $E$, since it receives one from $W$.

To prove uniqueness, let $a,b:E\rightrightarrows B$, and take their equalizer $j:V\to E$. Weak initiality of $W$ gives $t:W\to V$. Since $ejt$ is an endomorphism of $W$, the defining property of $e$ givesThe monomorphism $e$ cancels, yielding $jte=1_E$. Thus $j$ is both a monomorphism and a split epimorphism, hence an isomorphism. Its equalizer property now implies $a=b$. Therefore $E$ is initial.

Fix $C$. Small categorical limits in $(C\downarrow G)$ are obtained from the limits in $\mathcal{D}$: preservation by $G$ supplies the unique map from $C$ to the image of the limiting object. The comma category is locally small, and the solution family is a weakly initial set in it. The lemma supplies an initial object $(LC,{\eta }_C:C\to GLC)$.

For $k:C\to C^{\prime }$, initiality gives a unique arrow $Lk:LC\to L{C}^{\prime }$ satisfying $G(Lk){\eta }_C={\eta }_{C^{\prime }}k$. Uniqueness proves preservation of identities and composition, so $L$ is a functor. Initiality also gives natural bijectionswhich prove $L⊣G$.

Conversely, if $L⊣G$, the singleton ${\eta }_C:C\to GLC$ is a solution set by the adjunction bijection. To see that $G$ preserves a small limit $D={\lim }_j{D}_j$, use the adjunction and the limit property to obtain, naturally in $C$,This is precisely the universal property of the image limit cone. It proves the necessity of both hypotheses.

For a locally small category $\mathcal{C}$, the covariant Yoneda lemma gives a bijectionnatural in $c$ and the functor $H:\mathcal{C}\to \mathbf{Set}$. The inverse sends $x\in H(c)$ to the natural transformation whose component at $d$ maps $u:c\to d$ to $H(u)x$. Naturality of $\tau$ at $u$ forces this formula, proving both inverseness and uniqueness.

Write $\mathcal{E}=\int F$ for the category of elements of $F$. Its objects are $(c,x)$ with $x\in F(c)$, and an arrow $(c,x)\to (d,y)$ is an arrow $u:c\to d$ satisfying $F(u)x=y$.

Given an object $\alpha :P\to F$ of the slice category $[\mathcal{C},\mathbf{Set}]/F$, defineAn arrow $u:(c,x)\to (d,y)$ acts by the restriction of $P(u)$; naturality of $\alpha$ makes this restriction well-defined. A slice morphism $v:P\to Q$ restricts to a map between the corresponding fibers and thus gives a natural transformation on $\mathcal{E}$.

Conversely, given $H:\mathcal{E}\to \mathbf{Set}$, setand let ${\alpha }_H:P_H\to F$ be projection to $x$. The functor laws and naturality follow from those for $F$ and $H$. On morphisms use the maps on each summand. Taking a fiber of this disjoint union recovers $H(c,x)$ canonically. Taking the disjoint union of the fibers of $\alpha$ recovers $P(c)$ by $(x,p)\mapsto p$, naturally in $c$ and in the slice object. These are inverse natural isomorphisms, provingThis is the slice of a set-valued functor category construction.

Now assume $\mathcal{C}$ is a small category, so $\mathcal{E}$ is small. Define a diagram in a categoryFor $u:(c,x)\to (d,y)$ in $\mathcal{E}$, its reverse arrow acts by precomposition $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$. There is a cocone to $F$ with component $g\mapsto F(g)x$. To verify its colimit property directly, use pointwise colimits of sets. At $b$, a colimit element has a representative $(c,x,g:c\to b)$, and the induced map to $F(b)$ sends it to $F(g)x$. This map is surjective, since $z\in F(b)$ is represented by $(b,z,1_b)$. Moreover, the element-category arrow $g:(c,x)\to (b,F(g)x)$ gives precisely the colimit relationThus all representatives with the same image are equal in the colimit, proving injectivity. The resulting bijections are natural in $b$. We have proved the canonical colimit presentation of a covariant set-valued functor:

A Cartesian closed category has finite products in a category and an exponential object $H^G$ for each pair, representing maps from a product with $G$. In $[\mathcal{C},\mathbf{Set}]$, the terminal functor and binary products are computed pointwise. Define the exponential of covariant set-valued functors bySmallness makes this a set. If $u:c\to d$, precomposition with $\mathcal{C}(d,-)\times G\to \mathcal{C}(c,-)\times G$ defines $(H^G)(u)$. Functoriality is immediate from associativity of composition. Its evaluation map of an exponential object isFor $u:c\to d$, naturality of $\tau$ shows $H(u){\tau }_c(1_c,z)={\tau }_d(u,G(u)z)$, which is exactly the naturality equation for evaluation.

Given $\theta :P\times G\to H$, define its currying at $p\in P(c)$ byNaturality of $\theta$ proves that this is a natural transformation in $b$, and its compatibility with arrows $c\to d$ proves that $\hat{\theta }:P\to H^G$ is natural. Evaluation recovers $\theta$ by setting $f=1_c$. Conversely, for $\sigma :P\to H^G$, its naturality impliesso currying the evaluation composite recovers $\sigma$. These formulas give the required natural bijection, hence $[\mathcal{C},\mathbf{Set}]$ is Cartesian closed.

Finally, for every $F$ the small category $\mathcal{E}=\int F$ has a Cartesian closed functor category by the same construction. The slice equivalence above transports its terminal object, products and exponentials to $[\mathcal{C},\mathbf{Set}]/F$. The original functor category also has pointwise finite limits. Therefore it is locally Cartesian closed. Every construction here is explicit; no adjoint functor theorem is used.

The left adjoint to the morphism-set functor is the free category on independent arrows:Each copy has two objects, their identities, and one nonidentity arrow from the first object to the second. A functor $L(S)\to \mathcal{C}$ is specified by one arbitrary arrow of $\mathcal{C}$ for each $s$, since that arrow specifies the images of both objects as well. This proves the natural adjunction bijection with functions $S\to \mathrm{Mor}(\mathcal{C})$.

The induced monad on sets has $T(S)=S\times \{\text{source identity},\text{generator},\text{target identity}\}$. For a singleton $S$, its multiplication is a function from the nine-element set $T^{2}S$ to the three-element set $TS$, so it cannot be invertible. This adjunction is not idempotent.

For $L⊣U:\mathcal{D}\to \mathcal{C}$, let $T=UL$ be the induced monad. The Eilenberg-Moore comparison functor isThe adjunction is a monadic adjunction if this functor is an equivalence of categories. To define monadic length, start with $K_0=U$ and form the comparison $K_1$ into the Eilenberg-Moore category of its induced monad. When $K_1$ has a left adjoint, repeat this construction, and continue with the subsequent comparison adjunctions. The length is the least number of comparison steps after which the comparison is an equivalence; an equivalence has length zero and a monadic non-equivalence has length one. The successive left adjoints will all be explicit in this example.

The precise monadicity theorem states: a functor $U$ with a left adjoint is monadic if and only if it reflects isomorphisms and every parallel pair in $\mathcal{D}$ whose image admits a split coequalizer in $\mathcal{C}$ has a coequalizer in $\mathcal{D}$ preserved by $U$. Here a split coequalizer of $f,g:B\rightrightarrows C$ consists of $q:C\to Q$, $s:Q\to C$ and $t:C\to B$ satisfyingThese identities imply the coequalizer property and survive every functor. We shall identify the relevant algebra categories directly, which proves monadicity without having to check quotient constructions separately.

Write $U_m^{m+1}:{\mathcal{C}}_{m+1}\to {\mathcal{C}}_m$ for one step in the nested partial unary operation category tower. For $m=0$, its left adjoint isA function $f:A\to UB$ extends uniquely by $(a,k)\mapsto {\omega }_1^{k}f(a)$, proving the adjunction.

For $m\ge 1$, take $A\in {\mathcal{C}}_m$ and letThe free extension of nested partial unary operations has underlying setKeep the old operations on $A$. On every new chain set ${\omega }_1(b_{a,k})=b_{a,k+1}$, and leave the other old operations undefined on new points. Define the new operation by ${\omega }_{m+1}(a)=b_{a,0}$ for $a\in S_m(A)$, and nowhere else. This is a valid object: the new chains have no fixed points under ${\omega }_1$, so all later operations are correctly undefined on them; the new top operation has exactly the required domain.

If $f:A\to U_m^{m+1}B$ preserves the old operations, it carries $S_m(A)$ into $S_m(B)$. Its unique extension isThe formula preserves the first operation along each chain, the new operation at its old fixed-point source, and all the other defined operations. Conversely these requirements force the formula. This proves the one-step adjunction in every case.

Let $T_m=U_m^{m+1}{F}_m^{m+1}$. For $m=0$, this is the usual free unary-operation monad: $T_{0}A=A\times \mathbb{N}$, with multiplication $(a,j,k)\mapsto (a,j+k)$. An algebra action $h:A\times \mathbb{N}\to A$ obeys $h(a,0)=a$ and $h(h(a,j),k)=h(a,j+k)$. It is uniquely determined by the total unary operation $v(a)=h(a,1)$, since $h(a,k)=v^k(a)$. Algebra morphisms are exactly functions commuting with $v$. Thus ${\mathcal{C}}_0^{T_0}\cong {\mathcal{C}}_1$.

For $m\ge 1$, no new chain point belongs to $S_m(T_{m}A)$, so $S_m(T_{m}A)=S_m(A)$. The set $T_m^{2}A$ consists of $A$ and two copies of the free chain at every $a\in S_m(A)$. Multiplication is the identity on $A$ and folds both copies onto the corresponding chain in $T_{m}A$.

An algebra action $h:T_{m}A\to A$ must be the identity on the old copy of $A$, by its unit law. Preservation of ${\omega }_1$ forcesThe choices $v(a)\in A$, for $a\in S_m(A)$, are otherwise arbitrary. There are no higher old operations to preserve on new points. Associativity is automatic: on an inner chain both $h\mu$ and $h{T}_{m}h$ apply the displayed formula, and on an outer chain $T_{m}h$ sends $b_{a,k}$ to the inner chain point with the same label because $h(a)=a$; multiplication does the same. Therefore $h\mu =h{T}_{m}h$ on every point.

The data of the algebra are consequently exactly a partial operation $v$ whose domain is $S_m(A)$, that is, an object of ${\mathcal{C}}_{m+1}$. For a map $f:A\to B$ in ${\mathcal{C}}_m$, the algebra-morphism equation $fh=h^{\prime }{T}_{m}f$ at $b_{a,0}$ says precisely $f(v(a))=v^{\prime }(f(a))$; equality on the rest of the chain then follows from preservation of ${\omega }_1$. Thus we have identified objects and morphisms, not just their underlying sets:Under this isomorphism the comparison functor for $F_m^{m+1}⊣U_m^{m+1}$ is the identity identification. Hence every one-step forgetful adjunction is monadic.

Now take $n>m$. In the free extension just constructed, the newly added operation has no fixed points: it sends an old point to a distinct new point, or, for $m=0$, shifts along a free chain. Therefore all further operations can only be undefined. Adding them freely adds no more points. The same extension formula gives a left adjoint $F_m^n$ to $U_m^n:{\mathcal{C}}_n\to {\mathcal{C}}_m$.

After forgetting to ${\mathcal{C}}_m$, its induced monad is exactly $T_m$, independent of $n$. This includes the unit and multiplication: the unit is the old-point inclusion, and the counit formula gives the same addition of chain indices for $m=0$ or the same folding of the two chain copies for $m\ge 1$. Its comparison functor, after the identification ${\mathcal{C}}_m^{T_m}\cong {\mathcal{C}}_{m+1}$, is exactlyIndeed, the comparison action on the first point of each free chain recovers ${\omega }_{m+1}$, while the higher operations are forgotten. Thus each comparison step recovers exactly one additional operation. After $n$ steps from sets, the comparison is the identity of ${\mathcal{C}}_n$.

No earlier step is an equivalence. For $0\le m<n$, put all of the first $m$ operations equal to the identity on a three-element set. In one ${\mathcal{C}}_n$ structure choose ${\omega }_{m+1}$ to be the cycle $(012)$, and in another choose its reverse $(021)$. Both have no fixed points at this stage, so every higher operation is undefined. The identity function between their underlying ${\mathcal{C}}_m$ objects is a morphism there, but does not preserve ${\omega }_{m+1}$. Hence $U_m^n$ is not full, and cannot be an equivalence. This also applies for $m=0$, when there are no old operations. We conclude

A monoidal category consists of a category $\mathcal{C}$, a bifunctor $\otimes :\mathcal{C}\times \mathcal{C}\to \mathcal{C}$, a monoidal unit object $I$, and natural isomorphismsThe associator and unitors satisfy the pentagon identity for a monoidal categoryand the triangle identity for a monoidal categoryComposition is written from right to left. Naturality of these isomorphisms is part of the definition.

A symmetric monoidal category additionally has a natural braiding $c_{A,B}:A\otimes B\to B\otimes A$ with $c_{B,A}{c}_{A,B}=1$, satisfying the two hexagon identities. Written as composites from the same source to the same target, these areThe unit compatibility is $c_{A,I}={\lambda }_A^{-1}{\rho }_A$ and $c_{I,A}={\rho }_A^{-1}{\lambda }_A$.

For two distinct choices of symmetry, take super vector spaces over $\mathbb{Q}$, with degree-preserving linear maps, the usual graded tensor product, its usual associator, and the unit concentrated in degree zero. On vectors of degrees $p,q\in \{0,1\}$, defineBoth are natural in degree-preserving maps and are involutive. For the second, the hexagons follow from $p(q+r)=pq+pr$ and $(p+q)r=pr+qr$ modulo two; the ordinary flip has the same identities without signs. Degree zero of the unit gives unit compatibility. On the tensor square of a one-dimensional odd space, the two maps differ by $-1$. Thus the same monoidal structure can carry two different symmetries; the second uses the Koszul sign rule.

We prove the monoidal coherence theorem by a terminating normalization argument. Its precise formal assertion is that any two composites of associators, unitors, their inverses, and their tensor products with identities, between the same formal tensor expressions, are equal. The expressions must have the same ordered list of object letters after unit symbols are removed. This does not assert that different symmetries are equal, nor does it treat an accidental equality of actual objects as a new structural identification.

Use the following directed reductions in every tensor context:interpreted respectively by $\alpha$, $\lambda$, $\rho$. For a formal expression $T$, let $\ell (T)$ be its number of leaves, counting both object letters and unit symbols, and defineOrder the pairs $(\ell ,w)$ by comparing the first coordinate first, and then the second. A unit reduction decreases the first coordinate. An associativity reduction preserves the first coordinate and decreases the second by $\ell (A)>0$. The same calculation holds in any context, since associativity preserves the size of the rewritten subtree. Hence there is no infinite directed reduction path.

An irreducible expression has no unit next to another tensor factor and no composite left tensor factor. It is therefore the right-associated expression of its ordered nonunit letters, or the single unit $I$ if there are no such letters. In particular the normal form is uniquely determined by the ordered word.

We next show that every pair of first reductions has a commuting completion. Reductions in disjoint subexpressions commute by bifunctoriality of the monoidal tensor product. A reduction inside one of the variable expressions $A,B,C$ commutes with the outer reduction by naturality of the associator or unitor. The genuine overlaps are exactly the following cases.

For $(((A\otimes B)\otimes C)\otimes D)$, associating first at the root or first in its left subtree gives the two sides of the pentagon; their completions to $A\otimes (B\otimes (C\otimes D))$ commute.

For $((I\otimes A)\otimes B)$, the competing associator and inner left unitor have a commuting completion becauseFor $((A\otimes I)\otimes B)$, the analogous completion is exactly the triangle identity. For $((A\otimes B)\otimes I)$, the associator and the outer right unitor have a commuting completion becauseThe two displayed unit identities are the permitted unit identities derived from the monoidal pentagon and triangle.

The final overlap is $I\otimes I$, where both unitors apply. They agree as a consequence of the same identities. Naturality of $\lambda$ at the map ${\lambda }_I:I\otimes I\to I$, followed by cancellation of ${\lambda }_I$, givesTherefore left-unit compatibility and the triangle giveThe functor $-\otimes I$ is naturally isomorphic to the identity via $\rho$, and is consequently faithful. Thus ${\lambda }_I={\rho }_I$, so this overlap commutes too. There are no other nonvariable overlaps among the three reduction patterns.

Termination and these commuting completions imply equality of all normalization arrows, not merely uniqueness of the normal-form object. Here is the induction. Suppose every expression smaller than $T$ in the decreasing pair measure has a unique normalization arrow. Two normalization paths from $T$ start with arrows $r:T\to T_1$ and $s:T\to T_2$. The local completion gives directed paths $d_1:T_1\to V$, $d_2:T_2\to V$ with $d_{1}r=d_{2}s$. Append any normalization ${\nu }_V$ of $V$. By the induction hypothesis the tails of the original paths are ${\nu }_V{d}_1$ and ${\nu }_V{d}_2$. Their composites from $T$ are equal. If $T$ is already normal, its only directed normalization is the identity. This proves the induction.

Write ${\nu }_T:T\to N(T)$ for the resulting unique normalization. For every directed structural generator $g:S\to T$, including a generator in an arbitrary tensor context, uniqueness gives ${\nu }_{T}g={\nu }_S$. Since all structural generators are invertible, the corresponding equation holds for their inverses as well. Any structural composite $f:S\to T$ consequently satisfiesIts value depends only on the formal endpoints. Thus every structural diagram with the specified ordered word commutes, proving the monoidal coherence theorem.

Use the quotient convention for projective space. To a scheme $T$, associate the set of pairswhere $\mathcal{L}$ is a line bundle, modulo isomorphisms $\mathcal{L}\to {\mathcal{L}}^{\prime }$ carrying $q$ to $q^{\prime }$. Pullback defines the contravariant functor. On $T=\mathrm{Spec}A$, this means rank-one projective module quotients of $A^{n+1}$; the quotient need not be free. Thus merely listing generating tuples modulo multiplication by a unit would omit points over rings with nontrivial Picard group.

This functor is represented by ${\mathbb{P}}_{\mathbb{Z}}^n$. For each $i$, the locus where $q(e_i)$ generates $\mathcal{L}$ is open. These loci cover $T$, since a surjection onto a rank-one free stalk must have some coefficient a unit in its local ring. On this locus, trivialize by $q(e_i)$ and writeThese functions specify a map to the standard chart $U_i=\mathrm{Spec}\mathbb{Z}[t_j:j\ne i]$. Their transition rules are exactly those of projective space, so the chart maps glue. Conversely the universal quotient on projective space pulls back to the given pair, and the construction is natural and unique. This proves that projective space represents invertible quotients.

Write its universal exact sequence asA first-order change of the quotient, with target trivialized locally, is a map $\delta q:V\otimes \mathcal{O}\to \mathcal{O}(1)$. Changing the target trivialization by $1+\epsilon h$ changes $\delta q$ by $hq$. Restriction to $\mathcal{K}$ removes this ambiguity, and every local map $\mathcal{K}\to \mathcal{O}(1)$ extends because the displayed sequence splits locally. Consequently the tangent sheaf of a scheme isThis identification can also be seen on $U_i$: the first-order change of $t_j$ is $\delta q(e_j)-t_j\delta q(e_i)$ after setting $q(e_i)=1$. Thus these maps are precisely the coordinate tangent directions.

Apply $\mathcal{H}om(-,\mathcal{O}(1))$ to the universal sequence. Since its terms are locally free and it splits locally, this gives the Euler sequenceIt remains exact after any base change. Taking its dual of a sheaf over a field $k$ givesFor $n=1$, the first term has rank one; taking determinants yieldsExplicitly, for $u=t^{-1}$ on the overlap of the two affine charts, $du=-t^{-2}dt$, the same transition as $\mathcal{O}(-2)$ after changing one local frame's sign.

Both alternatives can be proved using the same smallness mechanism. We first establish the initial-object lemma for complete categories with a weakly initial set, then obtain the two adjoint functor existence theorems and their applications.

Let $\mathcal{E}$ be a locally small category with all small categorical limits and a weakly initial set $(W_i{)}_{i\in I}$. Its product in a category $W={\prod }_i{W}_i$ is weakly initial: given any object, choose a map from some $W_i$ and compose with the corresponding projection. Take the simultaneous equalizer $e:E\to W$ of every endomorphism of $W$ and its identity; local smallness makes this a small categorical limit. Then $E$ is also weakly initial. If $a,b:E\to X$, form their equalizer $j:Y\to E$ and choose $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, the construction of $e$ gives $ejte=e$. Cancelling the monomorphism $e$ yields $jte=1_E$. Thus $j$ is both a monomorphism and a split epimorphism, hence invertible, and $a=b$. There is therefore exactly one arrow from $E$ to every object: $E$ is initial.

The Freyd general adjoint functor theorem says that for a functor $U:\mathcal{A}\to \mathcal{B}$ between locally small categories, with $\mathcal{A}$ complete, a left adjoint exists exactly when $U$ preserves all small categorical limits and satisfies the solution-set condition. That condition requires, for each $B$, a set of arrows $u_i:B\to U{A}_i$ such that every $u:B\to UA$ factors as $U(f)u_i$ for some $i$ and $f:A_i\to A$.

For sufficiency, the comma category $(B\downarrow U)$ is locally small and complete: form a diagram's categorical limit in $\mathcal{A}$, and use categorical limit preservation to obtain its arrow from $B$. The solution set is a weakly initial set in this comma category. The lemma provides an initial object $(LB,{\eta }_B:B\to ULB)$. Its universal property givesFor $h:B\to B^{\prime }$, define $Lh$ by $U(Lh){\eta }_B={\eta }_{B^{\prime }}h$. Uniqueness proves the identity and composition laws and naturality of the displayed bijections, hence $L⊣U$. Conversely, an adjunction makes $U$ preserve categorical limits, since the corresponding hom-set functors turn a categorical limit cone into a categorical limit of sets. The single arrow ${\eta }_B$ is a solution set. This proves both directions of the theorem.

Now let $F:\mathcal{C}\to \mathcal{D}$ have small domain and codomain. The functor categories $[\mathcal{D},\mathbf{Set}]$ and $[\mathcal{C},\mathbf{Set}]$ are locally small, and their categorical limits are pointwise. Thus the precomposition functor $F^{\ast }$ preserves categorical limits and its domain is complete. To verify its solution-set condition, fix $X:\mathcal{C}\to \mathbf{Set}$ and an arrow $x:X\to F^{\ast }Y$. Let $Y_0\subseteq Y$ be the subfunctor generated by all elements $x_c(a)$, where $c\in \mathcal{C}$ and $a\in X(c)$. Explicitly,These sets are stable under every $Y(h)$. If $\kappa$ is an infinite cardinal bounding $|\mathrm{Mor}\mathcal{D}|$ and $\left|{\coprod }_{c}X(c)\right|$, every $Y_0(d)$ has cardinal at most $\kappa$. Replace each such set by a subset of a fixed set of size $\kappa$. There is only a set of choices of these subsets, of their action maps for the set of arrows of $\mathcal{D}$, and of maps from $X$ to their restrictions. These coded pairs form a solution set: $x$ factors through $F^{\ast }{Y}_0$ and then the inclusion $Y_0\hookrightarrow Y$. The general theorem now yieldsIts values may also be written ${\mathrm{Lan}}_{F}X(d)={\mathrm{colim}}_{(Fc\to d)}X(c)$, the usual left Kan extension formula.

For the other alternative, the categorical limit form of the Special adjoint functor theorem says that a limit-preserving functor $U:\mathcal{A}\to \mathcal{B}$ has a left adjoint if $\mathcal{A}$ is locally small, complete and well-powered, has a cogenerating set $(Q_i{)}_{i\in I}$, and $\mathcal{B}$ is locally small. A cogenerating set means that maps into its members distinguish any two parallel arrows. The dual theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family has a right adjoint. These conclusions are also necessary in their preservation clauses.

Here is a proof of the special theorem rather than an invocation of the general theorem without its required solution set. Given $(A,x:B\to UA)$, intersect all subobjects $m:M\hookrightarrow A$ through which $x$ factors after application of $U$. There are only a set of them by well-poweredness. Their intersection is a wide pullback, preserved by $U$, so gives a smallest supporting pair $(A_0,x_0)$. If $p,q:A_0\to Q_i$ satisfy $U(p)x_0=U(q)x_0$, their equalizer is a supporting subobject. Minimality makes that equalizer invertible, whence $p=q$. Consequentlyis injective. The cogenerating property makes the evaluation arrow from $A_0$ into the product in a category of the $Q_i$, one factor for each map $A_0\to Q_i$, a monic arrow. Its indexing set at each $i$ is identified with a subset of the fixed set $\mathcal{B}(B,U{Q}_i)$. Only a set of such products in a category can occur. Each has only a set of subobjects, and each possible subobject has only a set of arrows from $B$ into its image under $U$. These pairs give a weakly initial set in $(B\downarrow U)$, since every original pair receives the arrow from its minimal supporting pair. The complete-category lemma now gives the adjoint. Reversing all arrows proves the colimit form of the special theorem.

Apply that dual form to $F^{\ast }:[\mathcal{D},\mathbf{Set}]\to [\mathcal{C},\mathbf{Set}]$. Its domain is cocomplete and locally small, and $F^{\ast }$ preserves pointwise colimits. The set of covariant representable functors $\mathcal{D}(d,-)$ generates the domain: by the Yoneda lemma, maps from these representables probe every element at every object and hence distinguish distinct natural transformations. Finally, epimorphisms in this functor category are pointwise surjective. Indeed, their pointwise images form subfunctors; if an image is proper, the two inclusions into the pointwise pushout of two copies of the target over that image distinguish the original map, contradicting epicity. Quotients of a fixed functor are therefore specified by compatible equivalence relations on its sets of values, and form a set. Thus the domain is a well-copowered category. All special-theorem hypotheses hold, givingHere the Right Kan extension can be written ${\mathrm{Ran}}_{F}X(d)=\mathrm{Nat}(\mathcal{D}(d,F-),X)$. In particular both requested adjoints exist, although the two proofs use different smallness hypotheses.