Consider cospans of continuous maps $A\stackrel{f}{\rightarrow }C\stackrel{g}{\leftarrow }B$ and $A'\stackrel{f'}{\rightarrow }C'\stackrel{g'}{\leftarrow }B'$ along with maps $\alpha :A\to A'$, $\beta :B\to B'$ and $\gamma :C\to C'$ that make the corresponding diagram homotopy commutative.
It is well-known that if $\alpha $ and $\beta $ are $n$-equivalences and $\gamma $ is an $(n+1)$-equivalence, then the induced map between the corresponding homotopy pullbacks $A\times ^h_CB\to A'\times ^h_{C'}B'$ is an $n$-equivalence.
Is there a specific reference (book, paper) where I can find the proof of this fact? Or, at least, a clue for the proof?
By a q-equivalence I mean a map inducing isomorphisms of homotopy groups in degree <q and epimorphims in degree q, for every choice of base point.
