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 $n\ge1$.



Naturality of Moore-Postnikov systems
Working simplicially (in those days called "semi-simplicially") this is surely due to Moore, with details in unpublished 1956 lecture notes and in John C. Moore, Semi-simplicial complexes and Postnikov systems. 1958 Symposium internacional de topología algebraica International symposium on algebraic topology pp. 232–247. Earlier related work is in the 1954-55 Cartan Seminar "Algebres d'Eilenbeg-Maclane et homotopie". 


Taking $f$ and $g$ in the question to be Kan fibrations of simplicial sets, there is a brief treatment on pages 34-35 of my 1967 book "Simplicial objects in algebraic topology"(<http://www.math.uchicago.edu/~may/PAPERS1965.html>), where I define "the natural Postnikov system of the fibre space $(E,p,B)$". I don't remember whether or not I got that from my adviser's notes cited above, but I imagine I did.


