I am working through computing the homotopy of Thom spectra from Kochman's book. Let $A$ be a coalgebra over a field $k$, and let $M$ be a right $A$-comodule. Kochman constructs a coresolution $F$ of $M$ as follows: Define $F_0=M \otimes A$ and $\eta_0=\psi_M$ (the coaction of $A$ on $M$). Inductively, if $F_n$ and $\eta_n: K_n \rightarrow F_n$ have been defined, we define $K_{n+1}$ to be the cokernel of $\eta_n$. Then we define $F_{n+1}=K_{n+1} \otimes A$ and $\eta_{n+1}=\psi_{K_{n+1}}$ (the coaction of $A$ on $K_{n+1}$).
We then get short exact sequences $0 \rightarrow K_n \rightarrow F_n \rightarrow K_{n+1} \rightarrow 0$ Which we may splice together into a coresolution of $M$ by the $F_n$. Furthermore, this coresolution has the form F=F' \otimes A, which is important later when Kochman proves a change of rings proposition.
Question: In this construction, why does $F_n=K_n \otimes A$ have to be free?
$K_n$ is the cokernel of $\eta_n: K_n \rightarrow K_n \otimes A$, and it is not clear to me why this cokernel should be a direct sum of copies of $A$. I am not very comfortable working with comodules and coresolutions, so any help here will be greatly appreciated.