This is an exercise from Mcleary's book on Spectral sequence which I have been stuck with for some time. Let us recall what a Cartan-Eilenberg system is: IT consists of a module $H(p,q)$ for each pair of integers, $-\infty \leq p \leq q \leq \infty$ along with:
1) Homomorphisms $\eta:H(p',q') \rightarrow H(p,q)$ whenever $p \leq p'$, $q \leq q'$.
2) For $-\infty \leq p \leq q \leq r \leq \infty$ we have a connecting homomorphism $\delta:H(p,q) \rightarrow H(q,r)$.
3) $H(p,q) \rightarrow H(p,q)$ is the identity.
4) If $p \leq p' \leq p''$ and $q \leq q' \leq q''$ then the following diagram commutes $$\require{AMScd} \begin{CD} H(p'', q'') @>>> H(p,q) \\ @VVV @AAA \\ H(p',q') @= H(p',q') \end{CD}$$ 5) If $p \leq p'$, $q\leq q'$ and $r \leq r'$ then the following diagram commutes $$\require{AMScd} \begin{CD} H(p', q') @>>> H(q',r') \\ @VVV @AAA \\ H(p,q) @>>> H(q,r) \end{CD}$$ 6) For $-\infty \leq p \leq q \leq r \leq \infty$ , the following sequence is exact $$\cdots \rightarrow H(q,r) \rightarrow H(p,r) \rightarrow H(p,q) \xrightarrow{\delta} H(q,r) \rightarrow \cdots$$ 7) $H(-\infty,q)$ is the direct limit of the system $$H(q,q) \rightarrow H(q-1,q) \rightarrow H(q-2,q) \rightarrow \cdots$$
Then, Mcleary claims that we get a spectral sequence by letting $Z^p_r = im(H(p,p+r) \rightarrow H(p,p+1)$ , $B^p_r = im(H(p-r+1,p) \rightarrow H(p,p+1)$ and $E^p_r = Z^p_r/B^p_r$ . However, how does this give a spectral sequence? Mcleary defines a spectral sequence to be bigraded, first of all. Then I don't see any nice method to show that this gives a spectral sequence, so any help would be appreciated.