Example I proved from Weibel a year and a half ago. Just using as an example for the blog posts right now.
Let \( u: (C, d_C) \to (D, d_D) \) be a morphism of chain complexes with a family of \( R \)-module homomorphisms \( (u_i)_{i\in I} \). Then, \( u \) maps cycles to cycles and boundaries to boundaries, i.e. \( H_n(C) \to H_n(D) \).
We must show that \( u \) induces the maps (1) \( \ker d_{n,C} \to \ker d_{n,D} \) and (2) \( \Ima {d_{n+1, C}} \to \Ima {d_{n+1, D}} \).
For (1), observe that \( u_n: C_n \to D_n \) maps \( 0_C \to 0_D \). If \( k \in \ker d_{n,C} \), then \( u(k) \in \ker d_{n,D} \) since \( d_{n,D}(u(k)) = u_{n-1}(d_{n,C}(k)) = 0 \).
For (2), if \( j \in \Ima {d_{n+1,C}} \), there exists \( j^{-1} \in C_n \) with \( d(j^{-1}) = j \). By commutativity, \( u_n(j) = d_{n+1,D}(u_{n+1}(j)) \in \Ima {d_{n+1,D}} \).
Each \( H_n \) is a functor \( \mathbf{Ch(mod R)} \to \mathbf{mod R} \).
For any object \( \{C_n\} \) in \( \mathbf{Ch(mod R)} \), \( H_n({C_n}) \) is a right \( R \)-module as a quotient module. By Lemma 1, any morphism \( u_{C,D}: \{C\} \to \{D\} \) is mapped to a well-defined map \( H_n(u_{C,D}): H_n(C) \to H_n(D) \).