Where in the literature can I find a naturality statement for Moore-Postnikov towers of maps? Something like the following:

Let $f:X\to A$ and $g:Y\to B$ be maps of connected CW-complexes which both admit a Moore-Postnikov tower of principal fibrations. Then a commuting diagram $\require{AMScd}$ \begin{CD} X @>f>> A\\ @V \Phi V V @VV \phi V\\ Y @>>g> B \end{CD} (possibly with some extra conditions) induces maps $\Phi_n:X_n\to Y_n$ between the $n$-th stages of the towers of $f$ and $g$, for all $n\ge1$.