Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be a diagram in the category of topological spaces. If $g\circ f$ and one of $f,g$ are fibrations, can we conclude that so is the other?
In this question it is stated that if $f$ is surjective and $g\circ f$ and $f$ are fibrations then $g$ is a fibration. However, no proof is given there. Is this true? How do I prove it? I understand that the main difficulty is that an arbitrary map $W\to Y$ may not be lifted along $f$ to a map $W\to X$. I'm not sure whether this lifting property is true...
The motivation for this problem is that I want to show that $\mathbb{RP}^{2n+1}\to\mathbb{CP}^{n}$ and $\mathbb{CP}^{2n+1}\to\mathbb{HP}^n$ are fibrations. I know that there are fibrations (in fact fiber bundles) $\mathbb{S}^{2n+1}\to\mathbb{RP}^{2n+1}$ and $\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ and the factorization of these two maps gives the map $\mathbb{RP}^{2n+1}\to\mathbb{CP}^{n}$. I wonder if we could use the above property to deduce directly that $\mathbb{RP}^{2n+1}\to\mathbb{CP}^{n}$ is a fibration.