Turning a sequence of "zigzags" into a prespectrum Suppose that we are given a sequence of spaces $\{C_j\}$ and a family of diagrams
$$C_j \rightarrow D_j \xleftarrow{\sim} E_j \rightarrow \Omega C_{j+1}$$
Then, if we let
$$ F_j:= \operatorname{Holim}( C_j \rightarrow D_j \xleftarrow{\sim} E_j \rightarrow \Omega C_{j+1} \rightarrow \Omega D_{j+1} \xleftarrow{\sim} \Omega E_{j+1} \to \Omega^2 C_{j+2} \rightarrow \cdots)$$
Why is $F_j$ (weakly) equivalent to $C_j$?

The point is to replace this sequence of diagrams into a prespectrum, by using the fact that loops commute with homotopy limits and a morphism of diagrams in the right direction. I just can't see why we get the equivalence.
 A: We may compose the composable arrows in your diagram to obtain a zigzag of the form $$A_0 \rightarrow B_0 \leftarrow  A_1 \rightarrow B_1 \leftarrow \cdots$$ for which the directions of the arrows strictly alternate, which is initial in the original diagram.  So it suffices to give a formula to compute the (homotopy) limit of this reduced diagram.
We claim that $$\lim (A_0 \rightarrow B_0 \leftarrow  A_1 \rightarrow B_1 \leftarrow \cdots) \simeq \varprojlim_n (A_0 \times_{B_0} A_1 \times_{B_1} \cdots \times_{B_n} A_{n+1}).$$
Given this, we see that the limit in the question is equivalent to $$\varprojlim_n (C_j \times_{D_j} E_j \times_{\Omega D_{j+1}} \Omega E_{j+1} \times_{\Omega^2 D_{j+2}} \cdots \times_{\Omega^n D_{j+n}} \Omega^n E_{j+n}).$$  However, since we have $\Omega^k E_{j+k} \xrightarrow{\sim} \Omega^k D_{j+k},$ this iterated pullback collapses to just $C_j$ at each level.  So this inverse limit diagram is equivalent to a constant diagram at $C_j$, so the limit is just $C_j$.
We sketch a proof of the claim.  The essential ingredient is the technique to compute limits by decomposing the indexing diagram; see Corollary 4.2.3.10 in Higher Topos Theory for example.

*

*First, write the infinite zigzag diagram as an increasing union of finite zigzags starting at the left.  This shows that the limit can be computed as an inverse limit of limits of composable spans.

*Second, decompose the $n$-fold span into individual spans to show that the limit of iterated spans is computed by iterated pullbacks.

The formula in the claim follows by combining these two observations.
