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A collection of arrows is a two-sided ideal when arbitrary precomposition and postcomposition preserve it. If a finite category has split idempotent morphisms, every ideal $I$ satisfying $I^2=I$ is generated by identities belonging to $I$. Factor an arrow $f\in I$ repeatedly as a longer product of arrows of $I$. Two prefixes coincide, since there are only finitely many arrows. The intervening endomorphism $t\in I$ fixes that prefix; an idempotent power $e$ of $t$ still fixes it. Write $e=ir$, $ri=1_B$. Then $1_B=rei\in I$, and $f$ factors through $B$. Applied to the least covering sieves of a finite site, this proves rigid coverage.

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