Weibel exercise 1.1.2. the $n$th homology module is a functor from category Ch-Mod$(R)$ to Mod-$R$

Question

Ch-Mod$(R)$ is the category of $R$-module chain complexes. How do you turn a homology module into a functor?

Thanks for teaching.

Answer 1

Thanks to Pedro Tamaroff in a comment above.

I want to show that $H_n(\text{id}_{C}) = \text{id}_{H_n(C)}$.

$H_n(\text{id}_C) : H_n(C) \to H_n(C)$ is identity.

$H_n(f\circ g) = H_n(f) \circ H_n(g)$. Proof. Let $g_n' = H_n(g) : H_n(A) \to H_n(B) : z + B_n \to g_n(z) + B_n$. Clearly $f_n'\circ g_n' = H_n(f_n \circ g_n)$. Done