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An idempotent morphism satisfies $e^2=e$ and splits if $e=ir$ with $ri=1$. Form the relative Karoubi envelope $\mathcal{C}[\check{E}]$: its objects are $(A,e)$ for $e\in E$, andThe functor $A\mapsto (A,1_A)$ is full and faithful. Each $e\in E$ splits through $(A,e)$ using $e$ in both directions. If a functor $F:\mathcal{C}\to \mathcal{D}$ comes with splittings $F(e)=i_e{r}_e$, define its extension by $(A,e)\mapsto$ the splitting object and $u\mapsto r_{f}F(u)i_e$. This gives the required factorization, unique up to the unique compatible natural isomorphism.

A preorder $R$ is reflexive and transitive, so $R\circ R=R$: transitivity gives one inclusion and reflexivity the other. If it is an equivalence relation, the quotient relation $q:X\to X/R$ satisfies $q^{\mathsf{op}}q=R$ and $q{q}^{\mathsf{op}}=1$, so it splits in category of relations $\mathbf{Rel}$. Conversely, writing a split preorder as $R=ir$, $ri=1$ and using reflexivity and transitivity shows $R=R^{\mathsf{op}}$; hence it is an equivalence relation.

Let $B$ be the set of join-inaccessible elements of a completely algebraic lattice $A$, and putThen $ri=1_A$, while $r(S)\le a$ exactly when $S\subseteq i(a)$, so $r⊣i$. Join-inaccessibility gives $i({\bigvee }_j{a}_j)={\bigcup }_{j}i(a_j)$, so $i$ preserves all joins. Conversely, given such $r⊣i$ with $ri=1$, each $r\{b\}$ is join-inaccessible: if $r\{b\}\le \bigvee S$, adjunction and preservation of joins put $b$ in some $i(s)$, whence $r\{b\}\le s$. Finally,so $A$ is completely algebraic.

Under the supplied full embedding $\mathbf{Rel}\to \mathbf{CSLat}$, a set goes to its power-set lattice and a preorder goes to the associated idempotent. Splitting those idempotents gives exactly the retracts of power sets by adjoint join maps characterized above. Therefore $\mathbf{Rel}[\check{E}]$ is equivalent to the full subcategory of complete join-semilattices consisting of completely algebraic lattices.

A category is well-powered when the isomorphism classes of monomorphisms into each object form a set. For $P\in [\mathcal{C},\mathbf{Set}]$, every subobject is represented by a subfunctor $S$ with $S(A)\subseteq P(A)$. Since $\mathcal{C}$ is small, all choices lie in the set ${\prod }_A\mathcal{P}(P(A))$, and naturality merely cuts out a subset. Thus the functor category is well-powered. Quotients are similarly represented by compatible equivalence relations on the sets $P(A)$, so they form a subset of ${\prod }_A\mathcal{P}(P(A{)}^2)$; hence it is well-copowered.

A cocone under the identity diagram consists of maps $c_A:A\to L$ satisfying $c_{B}f=c_A$ for every $f:A\to B$. A terminal object supplies the unique such cocone and has the required universal property. Conversely, if $(L,c_A)$ is a colimit of the identity, both $1_L$ and $c_L$ mediate its cocone to itself, so uniqueness gives $c_L=1_L$. For any $f:A\to L$, cocone compatibility gives $f=c_{L}f=c_A$; thus there is exactly one arrow $A\to L$, and $L$ is terminal.

For $A\in \mathcal{C}$, let $\Omega (A)$ be the set of isomorphism classes of quotients of the representable $\mathcal{C}(A,-)$. This is a set by well-copoweredness. A map $u:A\to B$ sends a quotient of $\mathcal{C}(A,-)$ to the image quotient of the composite $\mathcal{C}(B,-)\to \mathcal{C}(A,-)$, making $\Omega$ a functor. For any functor $P$ and $x\in P(A)$, Yoneda gives $\mathcal{C}(A,-)\to P$; factor it as an epimorphism followed by a monomorphism and send $x$ to the resulting quotient class. These maps $P\to \Omega$ agree along every monomorphism. Conversely, a cone with apex $Q$ assigns to a quotient $q:\mathcal{C}(A,-)\twoheadrightarrow R$ the element obtained by applying its leg at $R$ to $q_A(1_A)$. Yoneda and epi-mono factorization show that this is well-defined and is the unique map $\Omega \to Q$. Therefore $\Omega$ is a local state classifier.

An object of $[\mathbb{Z},{\mathbf{Set}}_f]$ is a finite set with a permutation. For each $n>0$, let $A_n=\mathbb{Z}/n$ with trivial action and map a finite $\mathbb{Z}$-set $P$ to $A_n$ by sending every point to the length of its orbit modulo $n$. Equivariant injections preserve orbit lengths, so these maps form a cocone under the monomorphism subcategory; varying cyclic orbits shows that its legs are collectively surjective. If a local state classifier $L$ existed, its universal map onto every $A_n$ would be surjective because the universal legs are jointly epic. This would force the finite set $L$ to have at least $n$ elements for every $n$, a contradiction.