A morphism of chain complexes is a family of homs $u_n : C_n \to D_n $ such that $u_{n-1} d_n^{(C)} = d_n^{(D)} u_n$. Weibel's book says that cycles "get sent to cycles". To me that means that $u_n(Z_n^{(C)}) = Z_n^{(D)}$. I can show that lhs $\subset$ rhs easily: $x \in Z_n = \ker d_n^{(C)}$ means that lhs $= 0$ ($0$'s always map to $0$) and so $u_n(x) \in \ker d_n^{(D)}$ by the commuting formula. However, I can't show that if $y \in \ker d_n^{(D)}$ then there is a pre-image of $y$ in $\ker d_n^{(C)}$. So does she mean that $u$ maps cycles in $C$ into cycles in $D$ but not necessarily surjectively? Confused
Thanks!