Extensional quotient coalgebras

Given an endofunctor F of an arbitrary category, any maximal element of the lattice of congruence relations on an F-coalgebra (A, a) is called a coatomic congruence relation on (A, a). Besides, a coatomic congruence relation K is said to be factor split if the canonical homomor-phism ν : AK → A_∇A splits, where ∇_A is the largest congruence relation on (A, a). Assuming that F is a covarietor which preserves regular monos, we prove under suitable assumptions on the underlying category that, every quotient coalgebra can be made extensional by taking the regular quotient of an F-coalgebra with respect to a coatomic and not factor split congruence relation or its largest congruence relation.