In a series of papers, most of them authored or co-authored by H. Gonska, several authors investigated problems concerning the composition and decomposition of positive linear operators defined on spaces of functions. For example, given two operators with known properties, A and B, we can find the properties of the composed operator A∘B, such as the eigenstructure, the inverse, the Voronovskaja formula, and the second-order central moments. One motivation for studying composed operators is the possibility to obtain better rates of approximation and better Voronovskaja formulas. Our paper will address such problems involving compositions of some classical positive linear operators. We present general results as well as numerical experiments.