Injective Cogenerators in Acts Need Not Be Finitely Generated, Even for Coherent Monoids

We give a uniform construction of injective cogenerators in the category Acts of right acts over a monoid S via coinduction: the forgetful functor U admits a right adjoint. For any nonempty set X, the coinduced act is injective, and if |X| is at least 2 it is a cogenerator. We then show that for every infinite monoid S and every X with |X| is at least 2, the act is not finitely generated (by a cardinality bound on finitely generated subacts together with Cantor's theorem). Consequently, the act is not finitely presentable (equivalently, not finitely presented). In particular, this yields counterexamples over nontrivial right coherent monoids such as free monoids, which are known to be coherent.