Let $T$ be a well-founded poset and $k$ a field. Let $H: Ch_k \to Ho(Ch_k) = Gr_k$ be the homology functor and $\iota: Gr_k \to Ch_k$ be the canonical section which sends a graded vector space to the corresponding chain complex with zero differential.

**Proposition:** For any $F: T \to Ch_k$, $F$ and $\iota H(F)$ are quasi-isomorphic.

In particular, if $H(F) \cong H(G)$, then $F \simeq \iota H(F) \cong \iota H(G) \simeq G$.

**Proof:** In the Reedy = injective model structure on $Fun(T,Ch_k)$, fibrations and weak equivalences are levelwise, and an object is cofibrant (and thus bifibrant) if and only if each morphism of $T$ is carried to a monomorphism. We may assume that $F \in Fun(T,Ch_k)$ is cofibrant and prove the more specific claim that there is a quasi-isomorphism $F \to \iota H(F)$. By induction on the structure of $T$, we are reduced to the following:

**Lemma:** Let $X \to Y$ be a monomorphism in $Ch_k$. If $X \to \iota H(X)$ is a quasi-isomorphism, then there exists a quasi-isomorphism $Y \to \iota H(Y)$ forming a commutative square.

**Proof:** We may decompose $X \to Y$ into a series of cell attachments, and construct the map $Y \to \iota H(Y)$ by induction, one cell at a time. There are two cases. In the first case we have $Y = X \oplus k[n]$, in which case we have $H(Y) = H(X) \oplus k[n]$, and we extend using the identity on $k[n]$. In the second case we have $Y = X \cup_z w$ where $z$ is a cycle representing a nonzero homology class of $X$ and $w$ is a cell we are attaching to kill it. In this case we have $H(Y) = H(X)/v$, and we can extend via the zero map on $w$.