Summary: Let \(\mathcal M\) be a class of (mono)morphisms in a category \(\mathcal A\). To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair \((\mathcal{A,M})\). In this paper we take \(\mathcal A\) to be the category \(\mathbf{Act}\text{-}\mathbf S\) of \(S\)-acts, for a semigroup \(S\), and \(\mathcal M_{sd}\) to be the class of strongly \(s\)-dense monomorphisms and study the categorical properties, such as limits and colimits, of this class.