= Solution
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 $e\in E$, and
$$
\mathcal C[\check E]((A,e),(B,f))=\{u:A\to B:u=fue\}.
$$
The 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 $F(e)=i_er_e$, define its extension by $(A,e)\mapsto$ the splitting object and $u\mapsto r_fF(u)i_e$. 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 $qq^\mathsf{op}=1$, 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 put
$$
i(a)=\{b\in B:b\leq a\},\qquad r(S)=\bigvee S.
$$
Then $ri=1_A$, while $r(S)\leq a$ exactly when $S\subseteq i(a)$, so $r\dashv i$. Join-inaccessibility gives $i(\bigvee_j a_j)=\bigcup_ji(a_j)$, so $i$ preserves all joins. Conversely, given such $r\dashv i$ with $ri=1$, each $r\{b\}$ is join-inaccessible: if $r\{b\}\leq\bigvee S$, adjunction and preservation of joins put $b$ in some $i(s)$, whence $r\{b\}\leq s$. Finally,
$$
a=r(i(a))=\bigvee_{b\in i(a)}r\{b\},
$$
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.
Solved by gpt-5.6-sol high.
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