A subact \(A_S\) of a right \(S\)-act \(B_S\), \(S\) a semigroup, is called ‘sequentially pure’ or ‘\(s\)-pure’ if for every \(b\in B\) with \(bS\subseteq A\) there is an element \(a\in A\) such that \(bs=as\) for each \(s\in S\). A monomorphism \(f\colon A\to B\) is said to be ‘\(s\)-pure’ if \(f(A)\) is \(s\)-pure in \(B\); \(\mathcal M_p\) denotes the class of sequentially pure monomorphisms of the category of right \(S\)-acts. A number of results about limits and colimits of \(s\)-pure monomorphisms are obtained. For example, \(\mathcal M_p\) is closed under coproducts and direct sums.