Conditions such that split coequalizers are a symmetric notion

Question

Consider the notion of a split coequalizer (see the nLab for the definition). Note that the definition seems to be non-symmetric. Are there any conditions on the ambient category such that it becomes symmetric?

Answer 1

The definition of split coequaliser is essentially a transcriptions of that of an algebra for a monad, ie an object in the Eilengberg--Moore construction of the maximal category with an adjunction that yields the given monad.

Yes, it's asymmetric. The construction is awkward for other reasons. For example, the composite of two monadic adjunctions (ie of the Eilenberg-Moore kind) need not be monadic.

Sorry.

Any of the textbook accounts of monads will tell you about this, for example "Toposes, Triples and Theories" by Michael Barr and Charles Wells.