I'd be grateful for comments on the following question motivated by a problem in categorizing algebraic logic a la Tarski. Let C be a complete category with a factorization system (E,M). Given an object A\in C, let us call a pair (m,e) with m\in M, cod(m)=A and e\in E, dom(e)=A, {\it compatible}, m~e, if the square * --m--> A | | e' e | | v v *--m'--> A/e is pull-back (where (e',m') factorizes m;e ). In SET with the standard surjection-injection factorization, m~e iff for all a,b\in A, a\in A_m and e(a)=e(b) entail b\in A_m where A_m is the subset of A corresponding to m. Now let (e_i, i\in I) be a family of congruences compatible with some m:*--->A, e_i ~ m for all i\in I. The question is what properties of (C,E,M) are required to provide sup(e_i, i\in I) ~ m ? (the collection CongA of e:A-->*, e\in E is a meet-complete semilattice due to products, sup is join-via-meets in this lattice). If C is a category of finitary algebras over SET with the standard epi-mono factorization, then sup(e_i) ~ m always holds due to the finitary deduction property of taking sup in the congruence lattice: (a,b)\in sup(e_i) iff there exists a finite subfamily e_1,...,e_k and c_0,...,c_k \in A s.t. a=c_0, b=c_k and e_j(c_{j-1}) =e_j(c_j) for all j=1,...,k . Zinovy Diskin