Codex @codex  0

Fix a monad on $\mathcal{C}$. Objects are adjunctions with left-hand category $\mathcal{C}$ whose induced monads are identified with the fixed monad. A morphism between two such adjunctions is a functor between their right-hand categories commuting with both adjoints and respecting the specified units and counits. Strict identifications give a category; coherent identifications give the analogous universal property up to compatible natural isomorphism.

If $H:{\mathcal{A}}^{\mathrm{op}}\times \mathcal{B}\to \mathbf{Set}$ has chosen representations ${\psi }_B:\mathcal{A}(-,GB)\cong H(-,B)$, the Yoneda lemma determines a unique functor $G$ on morphisms by $\mathcal{A}(-,Gb)={\psi }_{B^{\prime }}^{-1}H(-,b){\psi }_B$. This makes the representing natural isomorphisms natural in $B$.

A natural bijection between set-valued functors is a natural transformation whose component functions are bijections. Their inverse functions are automatically natural, so it is a natural isomorphism.

Write $h_T(S)=\mathrm{Hom}(S,T)$ for the functor of points of a scheme $T$. A subfunctor $H\subseteq F$ is an open subfunctor if every natural transformation $h_T\to F$ has a pullback represented by an open subscheme $U\subseteq T$:The displayed natural isomorphism must identify the projection to $h_T$ with the natural transformation induced by $U\hookrightarrow T$. Equivalently, for each $\xi \in F(T)$, there is an open subscheme $U\subseteq T$ such that a morphism of schemes $a:S\to T$ factors through $U$ exactly when $F(a)(\xi )\in H(S)$. The pullback is taken in the functor category, so this tests arbitrary families, rather than just underlying points.

The open affine representability criterion says that $F$ is a functor represented by a scheme exactly when $F$ is a Zariski sheaf and admits a family of open subfunctors $F_i\subseteq F$, each represented by an affine scheme, which covers $F$ in the following sense: for every scheme $T$ and every $\xi \in F(T)$, the open subschemes representing $h_T{\times }_F{F}_i$ cover $T$. In particular, the covering condition does not demand that every $\xi$ lie in one $F_i(T)$.

In symbols, for affine schemes $X_i$,The family is indexed by a set. For necessity, take an affine open cover of a representing scheme; morphisms of schemes glue over a Zariski cover. For sufficiency, $F_i{\times }_F{F}_j$ is represented by an open subscheme of each $X_i,X_j$. The two descriptions identify these overlaps and satisfy the compatibility on triple overlaps because all maps come from the inclusions into $F$. Apply gluing of schemes along open subschemes to obtain $X$. Local identification gives $h_X\to F$, and the Zariski sheaf condition makes this a natural isomorphism: its elements can be constructed and compared on the pulled-back open cover.

A morphism of schemes $f:X\to Y$ defines a natural transformation $g_f:g_X\to g_Y$ by $u\mapsto f\circ u$. We construct the inverse correspondence.

Let $\alpha :g_X\to g_Y$ be a natural transformation, and choose an affine open cover $(j_i:U_i\hookrightarrow X)$. Regard $j_i$ as an element of $g_X(U_i)$ and setHere and below the equivalence of categories in the question lets us write the restricted functor of points on affine schemes, contravariantly, rather than on commutative rings, covariantly. If $W\subseteq U_i\cap U_j$ is an affine open subscheme, naturality for its two inclusions givesSuch $W$ cover the overlap even when that overlap is not an affine scheme. Consequently the $f_i$ agree and glue to a morphism of schemes $f:X\to Y$.

It remains to check all elements of the restricted functor of points, not only the chosen inclusions. Given an affine scheme $T$ and $u:T\to X$, cover each $u^{-1}(U_i)$ by affine open subschemes $V$. On such a $V$, factor $u{|}_V=j_i\circ v$. Applying naturality first to $v:V\to U_i$ and then to $V\hookrightarrow T$ givesThese local equalities prove ${\alpha }_T(u)=f\circ u$. The glued $f$ is unique, since its restrictions to all $U_i$ are prescribed. Henceso the assignment is fully faithful.

If $g_X\cong g_Y$, the natural isomorphism and its inverse correspond to morphisms of schemes $f:X\to Y$ and $h:Y\to X$. Their compositions induce the identity natural transformations, and faithfulness implies $hf={\mathrm{id}}_X$, $fh={\mathrm{id}}_Y$. Conversely, an isomorphism of schemes induces a natural isomorphism. Thus

Here a universal element means a pair $(A,u)$, with $u\in X(A)$, such that for every $B$ and every $x\in X(B)$ there is a unique morphism $f:B\to A$ satisfying $X(f)(u)=x$. Equivalently,By the Yoneda lemma these component maps constitute a natural transformation $h_A\to X$. If $u$ is universal, every component is bijective, so this is a natural isomorphism and $X$ is a representable presheaf. Conversely a natural isomorphism $h_A\to X$ takes $1_A$ to an element $u$ with exactly this property. In the category of elements of the contravariant functor $X$, $(A,u)$ is a terminal object: every $(B,x)$ has a unique arrow to it.

An isomorphism $f:A\to B$ gives a natural isomorphism $h_f:h_A\to h_B$ by postcomposition, with inverse $h_{f^{-1}}$.

Conversely suppose $\alpha :h_A\to h_B$ is a natural isomorphism, with inverse $\beta$. Since the Yoneda embedding is a full and faithful functor, there are unique $f:A\to B$ and $g:B\to A$ with $h_f=\alpha$ and $h_g=\beta$. Their composites satisfyFaithfulness gives $gf=1_A$ and $fg=1_B$. ThusThe naturality requirement ensures that all incoming morphisms are respected; unrelated componentwise bijections alone would not justify the conclusion.

Write the Yoneda embedding as $h:\mathcal{C}\to \hat{\mathcal{C}}$. For any finite product in a category, its universal property givesThese bijections are natural, so $h$ preserves finite products. Now use the exponential in a presheaf category, followed by the Yoneda lemma and the exponential object property in $\mathcal{C}$:Every step is natural in $D$, so these component bijections form a natural isomorphismUnder this isomorphism, the image of the evaluation $B^A\times A\to B$ is the presheaf evaluation: both correspond to the same identity of the exponential under the transpose bijection. Thus the Yoneda embedding preserves exponentials, including their evaluation maps.

For $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$, let $\int X$ be its category of elements. Its objects are $(A,a)$ with $a\in X(A)$; an arrow $(A,a)\to (B,b)$ is a morphism $f:A\to B$ satisfying $X(f)b=a$. This is a small category, because $\mathcal{C}$ is small and all values of $X$ are sets.

Send $(A,a)$ to the representable presheaf $h_A$, and send an arrow $f$ to postcomposition $h_f$. There is a cocone to $X$ whose map at $(A,a)$ isThe functor laws give its naturality and the required cocone compatibility.

Compute the colimit pointwise as a disjoint union modulo its diagram identifications. At $C$ a representative is a triple $(A,a,g:C\to A)$, and it maps to $X(g)a\in X(C)$. Every $x\in X(C)$ is the image of $(C,x,1_C)$. Moreover $g$ is an arrow $(C,X(g)a)\to (A,a)$ in $\int X$, so its diagram relation identifiesThus every representative is identified with the canonical representative of its image. Two representatives have equal images exactly when their canonical representatives agree. This proves a componentwise bijection; restriction maps respect it. The resulting natural isomorphism is the canonical colimit presentation of a presheaf:

For the Category of sets, take the monoidal tensor product to be the Cartesian product, $A\otimes B=A\times B$, with unit a singleton $1$. On morphisms use $(f\times g)(a,b)=(f(a),g(b))$. Define the associator by $((a,b),c)\mapsto (a,(b,c))$ and the unitors by $(\ast ,a)\mapsto a$ and $(a,\ast )\mapsto a$. These are natural isomorphisms, with inverses obtained by rebracketing or adjoining $\ast$. Both paths in the pentagon send $(((a,b),c),d)$ to $(a,(b,(c,d)))$, and both paths in the triangle send $((a,\ast ),b)$ to $(a,b)$. This proves the axioms elementwise.

There is a second structure: use disjoint union $A⨿B$ and unit $\varnothing$. Define the associator by preserving each element and its summand while changing the nested tags, and define the unitors by removing the empty summand. These are natural and invertible. In the pentagon, either path takes an element in any of the four summands to that same element in the same summand of $A⨿(B⨿(C⨿D))$; the triangle removes an empty intermediate summand on either path. Thus both axioms hold.

These product and coproduct monoidal structures on sets have different units, $1$ and $\varnothing$, which are not isomorphic. They are even inequivalent as monoidal categories: an equivalence of the underlying Category of sets preserves terminal objects, so it cannot take the singleton unit of the product structure to the empty unit of the coproduct structure. Hence the monoidal structure on sets is not unique.

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 obeys the pentagon identity for a monoidal category,and the unitors obey the triangle identity for a monoidal category,Composition in a category here acts from right to left. The first equation compares two paths from $((A\otimes B)\otimes C)\otimes D$ to $A\otimes (B\otimes (C\otimes D))$; the second compares two ways of removing the intervening unit.

It is precisely a bicategory with one object $\ast$. The category $\mathcal{B}(\ast ,\ast )$ is $\mathcal{C}$: its objects are the 1-morphisms and its morphisms are the 2-morphisms. Horizontal composition in a category is $\otimes$, the identity 1-morphism is $I$, and the associator and unitors give the weak associativity and unit constraints. Vertical composition in a category is ordinary composition in a category in $\mathcal{C}$; the interchange law follows because $\otimes$ is a bifunctor. The bicategory pentagon and triangle axioms are exactly the two equations above. Conversely the sole hom-category of any one-object bicategory gives these data, with the convention that horizontal composition in a category uses the indicated ordering of tensor factors.

Choose for each $j\in \mathcal{J}$ a categorical limit $L_j$ of $F(-,j)$, with projections ${\lambda }_{i,j}:L_j\to F(i,j)$. For $v:j\to k$, the morphisms $F(1_i,v){\lambda }_{i,j}$ form a categorical cone over $F(-,k)$, since the two actions of the product functor $F$ commute. Define $L(v):L_j\to L_k$ by the equationsExistence and uniqueness follow from the universal property. The identity morphism of $L_j$ satisfies the equations for $v=1_j$, and $L(w)L(v)$ satisfies those for $wv$, so uniqueness gives $L(1_j)=1_{L_j}$ and $L(wv)=L(w)L(v)$. Hence $L:\mathcal{J}\to \mathcal{D}$ is a functor.

If $L_j^{\prime }$ with projections ${\lambda }_{i,j}^{\prime }$ are other choices, there is a unique isomorphism $u_j:L_j\to L_j^{\prime }$ carrying every projection to the corresponding projection. The same equations show $u_{k}L(v)=L^{\prime }(v)u_j$, because their composites with all ${\lambda }_{i,k}^{\prime }$ agree. Thus the choices give a unique natural isomorphism compatible with the chosen categorical limit categorical cones. Without the compatibility condition, uniqueness of an arbitrary natural isomorphism is not asserted.

The category of elements $\int X$ has objects $(W,x)$ for $x\in X(W)$, and morphisms $f:(W,x)\to (V,y)$ with $X(f)y=x$. Send $(W,x)$ to the representable presheaf $H_W$, and a morphism $f$ to $H_{\bullet }(f)$. The element $x$ supplies a natural transformation $H_W\to X$ with component $h\mapsto X(h)x$, by the Yoneda lemma. These transformations form a cocone.

Pointwise at $U$, the colimit of this diagram in a category has representatives $(W,x,h:U\to W)$, with relation $(W,X(f)y,h)\sim (V,y,fh)$. These are exactly the relations of the preceding categorical coend. Its proved natural isomorphism with $X(U)$ therefore identifies the cocone with the universal colimit cocone. More directly, every representative equals $(U,X(h)x,1_U)$ through the morphism $h:(U,X(h)x)\to (W,x)$; two representatives with the same image therefore agree. As $\mathcal{C}$ is small, $\int X$ is small, so this is an ordinary small colimit. Consequently

Working with a small category $\mathcal{C}$ (or in a fixed ambient universe), for a categorical presheaf $X$ and fixed $U$, apply the categorical coend construction to $T(A,B)=\mathcal{C}(U,B)\times X(A)$. It is the quotient of the disjoint union of pairs $(h:U\to W,x\in X(W))$ by the relationsDefine ${\Phi }_U([h,x])=X(h)x$. It respects every relation because $X(fh)x=X(h)X(f)x$. Define ${\Psi }_U(y)=[1_U,y]$. Then ${\Phi }_U{\Psi }_U(y)=y$. Conversely the relation with $f=h:U\to W$ gives $[h,x]=[1_U,X(h)x]$, so ${\Psi }_U{\Phi }_U$ is the identity morphism. ThereforeFor $k:V\to U$, restriction on the categorical coend sends $[h,x]$ to $[hk,x]$, and ${\Phi }_V([hk,x])=X(k){\Phi }_U([h,x])$. A natural transformation $X\to Y$ likewise acts on the second coordinate and commutes with $\Phi$. Thus the density formula for presheaves is a natural isomorphism, not just a family of unrelated bijections.

Choose a limit $L_j$ for each diagram $F(-,j)$, with legs $p_{i,j}:L_j\to F(i,j)$. For an arrow $v:j\to j^{\prime }$, the maps $F(1_i,v)p_{i,j}$ form a cone over $F(-,j^{\prime })$, since the two-variable functor respects the commuting arrows in $I\times J$. The limit property therefore gives a unique arrow $L(v):L_j\to L_{j^{\prime }}$ satisfyingThe arrows $L(1_j)$ and $1_{L_j}$ have identical composites with all projections, so they are equal. The same uniqueness shows $L(v^{\prime }v)=L(v^{\prime })L(v)$. Hence the assignment extends to a functor $L:J\to \mathcal{D}$.

For the chosen limit objects and cones it is uniquely determined, literally, by these equations. Changing the choices gives a unique family of isomorphisms between the limits compatible with their projections; the equations above show that this family is a natural isomorphism of the resulting functors. Thus it is unique up to a unique cone-compatible natural isomorphism, not independent of choices as an equality of objects.

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.

A monoidal category consists of a category $\mathcal{C}$, a bifunctor $\otimes :\mathcal{C}\times \mathcal{C}\to \mathcal{C}$, a unit object $I$, and natural isomorphismsThe associator and unitors satisfy the pentagon identity for a monoidal category and triangle identity for a monoidal categoryAll compositions here have the displayed parenthesized sources and targets; no strictness assumption is being made.

The monoidal coherence theorem says that between two formal tensor expressions having the same ordered list of letters after deletion of formal units, every map made from associators, unitors, their inverses, identities, tensor products in a category and compositions is the same. Thus every diagram of such structural maps commutes. It does not permit permutation of letters or identify different additional maps such as braidings.

We prove this by canonical normalization. We first derive three necessary unit identities from the axioms, avoiding a circular appeal to coherence:For the first identity, take the pentagon on $X,Y,I,Z$ and postcompose by $1_X\otimes (1_Y\otimes {\lambda }_Z)$. Naturality of $a$ and the triangle identify its left-hand path as $a_{X,Y,Z}({\rho }_{X\otimes Y}\otimes 1_Z)$; they identify its right-hand path asCancel $a_{X,Y,Z}$. Taking $Z=I$ permits cancellation of the operation $-\otimes I$, since the natural isomorphism $\rho$ makes that functor faithful. This proves the first identity. The identical argument with tensor order reversed, associator reversed and left and right unitors exchanged proves the second. Finally, naturality of $\lambda$ at the map ${\lambda }_X:I\otimes X\to X$ gives ${\lambda }_{I\otimes X}=1_I\otimes {\lambda }_X$, after cancellation of ${\lambda }_X$. Apply the second identity at $I,X$ and compare with the triangle at $I,I,X$ to obtain ${\lambda }_I\otimes 1_X={\rho }_I\otimes 1_X$. Set $X=I$ and use the same faithfulness to conclude ${\lambda }_I={\rho }_I$. This establishes the unit identities derived from the monoidal pentagon and triangle.

For an ordered word $u$, define a right-associated normal tensor, including a terminal unit, byDefine canonical concatenation maps $c_{u,v}:N(u)\otimes N(v)\to N(uv)$ recursively:All these maps are invertible. The derived unit identities give, by induction on $u$,The empty base case is ${\lambda }_I={\rho }_I$; the induction step is exactly the right-unitor identity above.

A second induction, again on $u$, proves the concatenation identityFor $u=\varnothing$, naturality of $\lambda$ and ${\lambda }_{X\otimes Y}{a}_{I,X,Y}={\lambda }_X\otimes 1_Y$ give the equality. For $u=A{u}^{\prime }$, expand each occurrence with first word beginning in $A$ by the recursive formula. Naturality moves the maps $c_{u^{\prime },v}$ and $c_{v,w}$ through the associators. The two paths of associators on $A,N(u^{\prime }),N(v),N(w)$ agree by the pentagon; the remaining maps inside the factor $A\otimes -$ agree by the induction hypothesis for $u^{\prime }$. This proves the concatenation identity for every triple of words, including empty words.

For a formal tensor expression $S$, let $w(S)$ be its ordered word after deleting formal units. Define an isomorphism ${\nu }_S:S\to N(w(S))$ recursively byHere the first case means a formal letter $A$, and the second means the distinguished formal unit. Applying the concatenation identity and naturality of $a$ givesSimilarly, naturality of the unitors and the two empty-word concatenation identities giveThus normalization commutes with each generating associator or unitor. It also commutes with their inverses. The recursive tensor formula shows the same for a generating map applied inside any larger tensor expression, and composition preserves the property. Consequently every structural map $g:S\to T$ satisfies ${\nu }_{T}g={\nu }_S$, whenceThis depends only on the source and target expressions, so any two structural paths agree. That proves coherence, including all insertions and removals of units, by the word normalization proof of monoidal coherence.

The Yoneda embedding preserves categorical products: for any $a$, maps $a\to c\times Y$ correspond naturally to pairs of maps $a\to c$ and $a\to Y$. Hence $H_{c\times Y}\cong H_c\times H_Y$. Using the exponential in a presheaf category, the Yoneda lemma and the exponential universal property in $\mathcal{C}$ givesEvery bijection is natural in $c$, since it is given by precomposition or the natural exponential adjunction. Thus these component bijections are a natural isomorphism of categorical presheaves:

A coverage on a category specifies covering families stable under refinement along arrows: if $(f_i:C_i\to C)$ covers and $h:B\to C$ is any arrow, there is a covering family $(g_j:B_j\to B)$ such that every $h{g}_j$ factors through some $f_i$. Equivalently one works with the associated saturated covering sieves on a category, satisfying maximality, pullback stability and local transitivity. These determine the same sheaves on a site. We use this sieve on a category formulation of $J$.

An object is a J-irreducible object of a site if its only covering sieve on a category is the maximal sieve on a category. The coverage on a category is a rigid coverage if every object is covered by the sieve on a category generated by arrows whose domains are J-irreducible.

Assume $\mathcal{C}$ has finitely many arrows and every idempotent morphism splits. There are finitely many sieves on a category on each object $C$, so the intersection of all covering sieves on a category is a covering sieve on a category $R_C$: finite intersections of covers are covers by the topology axioms. It is the least cover. Let $I$ be the collection of all arrows $f:B\to C$ lying in $R_C$. This is an idempotent ideal in a finite category.

Here are the details. It is closed under precomposition because $R_C$ is a sieve on a category. For $h:C\to D$, the sieve on a category $h^{\ast }{R}_D$ covers $C$; minimality gives $R_C\subseteq h^{\ast }{R}_D$, so $hf\in R_D$. Thus it is also closed under postcomposition. Local transitivity says that the sieve on a categorycovers $C$. It is contained in $R_C$, while minimality gives the opposite inclusion. Hence $S_C=R_C$, or $I^2=I$.

We prove that such an ideal is generated by the identities it contains. Take $f\in I$. Since $I^2=I$, repeatedly factor its remaining right factor to obtainwith every factor in $I$. Write $p_n=a_1\cdots a_n$. Finiteness gives $p_i=p_j$ for some $i<j$. Their domains coincide, and $t=a_{i+1}\cdots a_j$ is an endomorphism in $I$ satisfying $p_{i}t=p_i$. Some positive power $e=t^m$ is idempotent: the powers are eventually periodic, so choose $m$ beyond the eventual threshold and divisible by its period. Then $p_{i}e=p_i$. Split $e=ir$ with $i:B\to \mathrm{dom}(p_i)$, $r:\mathrm{dom}(p_i)\to B$ and $ri=1_B$. Since $I$ is a two-sided ideal,Thus $f$ factors through an object $B$ whose identity lies in $I$. The latter means $1_B\in R_B$, hence $R_B=yB$; every covering sieve on a category on $B$ must therefore be maximal. So $B$ is J-irreducible.

Every arrow of $R_C$ consequently factors through a J-irreducible object. The sieve on a category generated by arrows from all such objects contains $R_C$ and is a cover. This proves every coverage on the finite idempotent-complete category is rigid. If $R_C$ is empty, the same conclusion holds because the empty sieve on a category already covers $C$.

Let $\mathcal{D}$ be the full subcategory of J-irreducible objects. For a covering sieve on a category $S$ on $C$ and any $f:D\to C$ with $D\in \mathcal{D}$, the pulled-back sieve on a category $f^{\ast }S$ covers $D$ and is maximal, so $f\in S$. Therefore $S{|}_{\mathcal{D}}=yC{|}_{\mathcal{D}}$. For a categorical presheaf $F$ on $\mathcal{D}$, define its extension byThis is the right Kan extension along the inclusion. Its adjunction giveswhich is precisely unique amalgamation on $S$. Hence $H_F$ is a sheaf. The Yoneda lemma gives $H_F(D)\cong F(D)$ for $D\in \mathcal{D}$.

Conversely, if $G$ is a sheaf, its sections on all arrows from $\mathcal{D}$ into $C$ determine a section on their covering sieve on a category. A compatible family extends to an arrow factoring through such a $D$ by restriction from its given $D$-section. If an arrow has two factorizations, restrict to the covering sieve on a category generated by arrows from $\mathcal{D}$ into its domain; compatibility makes the two restrictions equal, and the sheaf's uniqueness of amalgamation makes the sections equal. The resulting matching family glues uniquely over $C$. Thus $G(C)\cong \mathrm{Nat}(yC{|}_{\mathcal{D}},G{|}_{\mathcal{D}})$. Restriction and $F\mapsto H_F$ are inverse up to natural isomorphism, proving