In this paper, we introduce and analyze the class of $(a,k)$-regularized $(C_{1},C_{2})$-existence and uniqueness families in the setting of sequentially complete locally convex spaces. The classes of $(a,k)$-regularized $C_{1}$-existence families and $(a,k)$-regularized $C_{2}$-uniqueness families are also defined and considered. The subordination principle as well as many other structural characterizations of (local) exponentially equicontinuous $(a,k)$-regularized $(C_{1},C_{2})$-existence and uniqueness families are proved.