In the survey paper "Cartan's Constructions, the homology of $K(\pi,n)$'s, and some later developments" by J. C. Moore ( http://www.numdam.org/item/AST_1976__32-33__173_0/ ) he states a theorem:
Let $A$ be a DG-algebra over a ring $R$. Let $M$ be a split-#-projective $A$-module. Let $f : M'\to M''$ be a morphism of $A$-modules, which is a homotopy equivalence of the underlying chain complexes of $R$-modules, and which is a split epimorphism of graded $R$-modules. Then any $g: M\to M''$ lifts along $f$, and the lift is unique up to a homotopy respecting the action of $A$.
He says that Cartan proved this in the case that $M''=R$ but I regard it as an important insight that it can be extended more generally to the case $M''\neq R$. Who is responsible for the more general theorem stated here? It could obviously be due to Moore himself but did Moore write this down anywhere prior to this survey paper presented a decade after the results of the Cartan seminar? Is there a proof of it written down somewhere?
(The implicit notions of cofibrant/fibration/weak equivalence suggested in the statement of theorem do form a model structure on DG $A$-modules but this was not proven until 2016, see http://www.math.uchicago.edu/~may/PAPERS/BMRSix.pdf)
