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(An elementary version of) 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: $$[F,G]\cong \mathsf{Nat}(H_0F,H_0G).$$

A universal $\delta$-functor $F_\bullet:\mathsf{A}\rightarrow \mathsf{B}$ is a $\delta$-functor for which giving a morphism $\tau_\bullet:F_\bullet\rightarrow G_\bullet$ of $\delta$-functors is equivalent to merely giving $\tau_0$. In other words (assuming $\delta$-functors from $\mathsf A$ to $\mathsf B$ and their morphisms constitute a category $\delta$, we have a bijection $$\mathsf{Hom}_\delta(F_\bullet,G_\bullet)\cong \mathsf{Nat}(F_0,G_0).$$

For both concepts, "extending from the zeroth level" seems to play a central role. I'm trying to understand precisely what is the formal (and informal) relationship between them.

This MO question may be relevant, but I'm unable to understand the answers fully, so if they help, please explain them in detail.

What is the big picture here?

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