If M, A are two differential graded complexes over a commutative ring R with the following data, $$(M,d_M) \overset{\Delta}{\longrightarrow} (A,d_A)$$ $$(A,d_A) \overset{f}{\longrightarrow} (M,d_M)$$ $$A \overset{\phi}{\longrightarrow} A $$ with the following conditions, $f \Delta=1_{M}, \Delta f=1_{A}-(d_{A} \phi+\phi d_{A}) , \phi \Delta=0, f \phi=0 $. Such a data is called Strong Deformation Retract Data(SDR).
In this setup, suppose t is a perturbation of $d_{A}$ i.e ($d_A+t$ )is a new differential in A which amounts to saying t satisfies $ [d_A,t]=0 , t^{2}=0$. Then Homotopy Transfer lemma allows you to define a new differential on $(M,\Delta_{\infty})$ so that $(M,\Delta_{\infty}) , (A,d_A+t)$ are SDR pair.
Such a solution can be formally written as described in this paper


I wanted to know if there is any Homotopy Transfer Lemma without the hypothesis $f\Delta=1$. What if there is another map $\psi: M \rightarrow M $ satisfying $f\Delta=1_{M} -(d_{M}\psi+\psi d_{M})$? Can we still expect such a lemma? If yes, can we write down an explicit formula for the new differential as before? Please point me to the right reference. Thank you!


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