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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 $n\ge1$.

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    $\begingroup$ This was surely known to Moore, who only seems to have considered the case A=B=*, and it's mentioned in passing in more modern references (like Goerss-Jardine). One reference that explicitly creates a functorial fiberwise localization functor, in great generality, is Dror Farjoun's book (see Example E.1 and section F of Chapter 1 in "cellular spaces, null spaces, and homotopy localization"). etc. etc... $\endgroup$ – Dylan Wilson Mar 6 '18 at 21:58
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    $\begingroup$ But are you interested more in having the earliest reference where it "basically" appears, or just a true statement, no matter how modern the reference? $\endgroup$ – Dylan Wilson Mar 6 '18 at 22:00
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Working simplicially (in those days called "semi-simplicially") this is surely due to Moore, with details in unpublished 1956 lecture notes and in John C. Moore, Semi-simplicial complexes and Postnikov systems. 1958 Symposium internacional de topología algebraica International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.

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