A recurrent formula of $A_{\mathrm{\infty }}$-quasi inverses of dg-natural transformations between dg-lifts of derived functors

A dg-natural transformation between dg-functors is called an objectwise homotopy equivalence if its induced morphism on each object admits a homotopy inverse. In general an objectwise homotopy equivalence does not have a dg-inverse but has an $A_{\mathrm{\infty }}$ quasi-inverse. In this note we give a recurrent formula of the $A_{\mathrm{\infty }}$ quasi-inverse. This result is useful in studying the compositions of dg-lifts of derived functors of schemes.