In this paper, convenient properties of stratified L-convergence tower spaces are investigated. Firstly, it is shown that the construct of stratified L-convergence tower spaces and the construct of stratified L-Kent convergence tower spaces are strong topological universes. Secondly, the concepts of symmetric stratified L-Kent convergence tower spaces and complete stratified L-filter tower spaces are introduced and it is proved that the resulting constructs are isomorphic and they are strongly Cartesian closed