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

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