Much as I try to get out of the habit of finding counterexamples to things, here, for what it's worth, is a category with a map satisfying Mike Barr's condition but which is not a coequaliser. * ====> * <==== * | \ / | \ / | | \ / f \ / | | X | X | | / \ | / \ | V V V V V V V * * * or, more prettily, $$\begin{diagram} \bullet&\pile{\rArr\\\rArr}&A&\pile{\rArr\\\rArr}&\bullet\ \dArr&\SE\SW&\dArr~f&\SE\SW&\dArr\ \bullet&&B&&\bullet \end{diagram}$$ ("=" above means a parallel pair. Believe it or not, I typed that without an editor!) A mono satisfying Mike's condition is invertible. Can't see how to relate it to orthogonality (in the sense of factorisation systems) with monos. What I had in mind was that someone may have used coequalisers in distributive categories to model "while". Only those coequalisers which, as relations and hence directed graphs (the node set being the target of the parallel pair), are directed arise in this way. Paul Taylor =========================================================================