Universal monoid actions: the power of freedom

Motivated by a recent work of Balcerzak and Kania [Proc. Amer. Math. Soc. 151 (2023) 3737–3742], we show that every countable monoid has a universal action on the free object over a countable infinite set. This is a general result concerning concrete categories with a left adjoint (free) functor. On the way, we introduce an abstract concept of ``being generated by a set". At the same time we obtain a simpler proof of the result of Darji and Matheron concerning a surjectively universal operator on the classical Banach space ${\mathrm{\ell }}_1$ of summable sequences.