Can we always find the smallest equivalence relation satisfying some conditions in an algebraic category? If so, do we always know how?
I'm currently dealing with $\bf Mon$ so I have very few operations at my disposal. What I understand is that it is unique, since it's determined by its equivalence classes. It exists since we always have the trivial eq-rel $M\times M$ which equates all elements with each other.
I'm dealing with coequalizers and it feels rather vague to just call upon this, although I did the same with $\bf Set$.
Further here we have various constraints, because it must respect multiplication and all equivalence classes must commute with the identity class. But maybe this is ensured by the monoid morphisms.
Anyway, I know I'm only half coherent right now and I apologise for that. Hope somebody can make sense of my question and put me at ease. Thanks either way for taking the time to read this far.