[ I asked the same question on stackexchange but attracted little attention. Besides, I made some progress after I posted it. So I decided to move it here. ]

Consider path-connected CW-complexes $A$, $B$, $C$, $X$.
I would like to show the equivalence of the following two conditions.

(i) They fit into fibrations $A\to X\to Y_1$ and $B\to Y_1\to C$ for some $Y_1$.

(ii) They fit into fibrations $A\to Y_2\to B$ and $Y_2\to X\to C$ for some $Y_2$.

Everything is taken up to homotopy equivalence.

I can prove (i) $\Rightarrow$ (ii) as follows. It's not hard to show that we can always find two fibrations $X\stackrel{f}{\to}Y_1$ and $Y_1\stackrel{g}{\to}C$ that are homotopy equivalent to those fibrations in (i) but the two $Y_1$'s here are precisely the same space. Here $X$, $Y_1$, and $C$ are homotopy equivalent to but perhaps different from those spaces above, although I just abuse the same symbols. Then $X\stackrel{h=gf}{\longrightarrow}C$ is also a fibration. Let's pick up an arbitrary $c\in C$. Then $h^{-1}c\stackrel{f}{\to}g^{-1}c$ is clearly again a fibration, whose fiber is the same as a fiber of $X\stackrel{f}{\to}Y_1$ thus homotopy equivalent to $A$. We also note that $g^{-1}c$ is a fiber of $Y_1\stackrel{g}{\to}C$ thus homotopy equivalent to $B$. Therefore, letting $Y_2=h^{-1}c$ proves (ii).

But I found difficulties in proving (ii) $\Rightarrow$ (i). My strategy is to rephrase (ii) in another form similar to (i), i.e.,

(ii) They fit into fibrations $\Omega B\to A\to Y_2$ and $\Omega C\to Y_2\to X$ for some $Y_2$.

Then according to (i) $\Rightarrow$ (ii), we can prove (ii) $\Rightarrow$ (iii) where (iii) is the following.

(iii) They fit into fibrations $\Omega B\to Y_3\to \Omega C$ and $Y_3\to A\to X$ for some $Y_3$.

If $Y_3$ is deloopable, letting $Y_1=BY_3$ proves (i). But I don't know how to argue $Y_3$'s deloopability. I also tried to construct counterexamples, but I still failed. During those trials, I strongly felt that (ii) prohibits me to destroy $Y_3$'s deloopability.

**Question:** Could anyone help me complete my proof?