Let $f:A\to B$ be a monoid homomorphism. Where can I find an explicit description of the its cokernel? Are there any books on this topic?
If anyone cares, here's my motivation. In the category of groups, the cokernel of the kernel of a group homomorphism $f$ is the quotient of the domain by the kernel, which is comprised of the cosets of the kernel. The first isomorphism theorem says this quotient is isomorphic to the image. This makes sense because the multiplicative kernel action has strongly connected components (because of the existence of inverses), so the cosets of the kernel are the fibers.
For monoids there's no first isomorphism theorem because the kernel is largely uninformative. However, some monoid epimorphisms are known to be the cokernels of their kernels (namely Schreier split monoid epimorphisms), and I would like to see what this means concretely.
Added. By the cokernel of $f$, I mean the universal arrow which postcomposes with $f$ to give the trivial homomorphism. Sorry for not including this from the start, I just thought there's no risk of ambiguity.