Profinite Heyting algebras, and partitions of image-finite posets under open maps

We are interested in developing a theory of quotient objects (i.e. partitions) in sufficiently general categories related to non-classical propositional logics. Our objective, is to obtain a notion of partition for an image-finite poset P that (i) dualises subobjects in the category of profinite Heyting algebras and their complete homomorphisms, and (ii) can be directly described in terms of 'parts' of P, without reference to epic arrows in the category of image-finite posets and open maps.