In example (viii) of section 1.3.2 of Category Theory in Context, the $n$-cycle functor is defined on objects as
\begin{align} Z_n: \mathbf{Ch_R}&\to\mathbf{Mod_R} \\ C_\bullet&\mapsto \ker(d:C_{n+1}\to C_n) \end{align}
Considering all degrees simultaneously, the cycle functors assemble into a functor $Z_∗ : \mathbf{Ch_R} → \mathbf{GrMod_R}$ from the category of chain complexes to the category of graded $R$-modules.
It was easy to understand how $Z_n$ behaves, but I failed to define $Z_∗$. Even though $R$-modules $Z_nC_\bullet$ can be selected out of $C_\bullet$, they have to be assembled in a single graded $R$-module. How is this done?
What is a graded $R$-module? It's an $R$-module $X$ equipped with a splitting $X \cong \bigoplus_n X_n$. Here you can just define $Z_*C = \bigoplus Z_nC$, and then it is an $R$-module equipped with an obvious splitting. (As others have said, there is no multiplication here, unless there was one on $C$ compatible with the differential.)