This short paper is about the behavior of the cellular automaton (CA) obtained by the composition of two or more cellular automata. The general question we aim to face is: what is the relationship between a certain property of the CA obtained by the composition and the same (or other) property of each single CA appearing in the composition? We show that if P is one of the main basic and set-theoretic properties, the CA resulting from the composition has a certain property P iff both CA components have P, while this equivalence generally no longer holds when P is one of the main dynamical properties. Some important questions naturally arise from these results.