The fundamental lemma of homological algebra discusses the extension of arrows to chain maps from a projective to an arbitrary resolution, and the uniqueness-up-to-homotopy of such an extension. Indeed, if $\mathsf{A}$ is an abelian category and $P_\bullet, Q_\bullet$ are respectively a projective and an arbitrary resolution of objects $A,B$, then the fundamental lemma says $$\mathsf{Hom}_\mathsf{A}(A,B)\cong [P_\bullet,Q_\bullet].$$
The acyclic model theorem seems to go in much the same spirit. The main difference seems to be the the acyclic model theorem has a notion of freeness, while the fundamental lemma employs projectivity. Indeed, the acyclic models theorem more-or-less says that natural transformations between the zeroth homology of a free functor taking values in $\mathsf{Ch}^+_\bullet(\mathsf A)$ to the zeroth homology of a parallel acyclic functor are in bijection with (chain) homotopy classes of natural transformations: $$\mathsf{Nat}(H_0F,H_0G)\cong [F,G].$$
Is the fundamental lemma of homological algebra a special case of (some version of) the acyclic model theorem? What is the big picture here?
