On the homotopy transfer of $A_{\mathrm{\infty }}$ structures

The present article is devoted to the study of transfers for $A_{\mathrm{\infty }}$ structures, their maps and homotopies, as developed in \cite{Markl06}. In particular, we supply the proofs of claims formulated therein and provide their extension by comparing them with the former approach based on the homological perturbation lemma.