I assume that by "regular" you mean "regular mono". If that is so then I think that your X is "difunctional jointly monic pair". Aurelio, Christina and company have recently looked at difunctional relations in their work on Mal'cev matters but I do not have an explicit reference for you. The X above is an ugly mouthfull but it makes sense in that for the most "decent" of categories, toposes, every mono is "regular" suggesting a usualness or normalness about such monos relative to the paradigm, one to one function_but difunctionality is not usual. To be precise, let X<---R--->A be a jointly monic pair in SET then R is a pullback of some cospan if and only if, viewed as a relation, R is "transitive" in the following sense: aRx and bRx and bRy implies aRy . This interpretation is of course possible in much greater generality than SET. Best regards, Richard ==========================================================================