Compatibility of Definitions of Universal G-bundles In the book Fibre Bundles by Husemoller, universal G-bundles are introduced as bundles over a homotopy type $BG$, for which the cofunctor $[-,BG]\rightarrow k_G(-)$ is a natural isomorphism. 
Contrary to this, tom Dieck defines universal G-bundles as those for which the total space EG is terminal for some homotopy category, i.e. for all numberable free G-spaces $E$ there is a G-map $E\rightarrow EG$, unique up to G-homotopy.
How does the first definition imply the second? My aim is to get a unique homotopy type to show that any total space of a universal bundle as defined in the first way is contractible without restricting to CW complexes.
I have already unsuccessfully asked this question here https://math.stackexchange.com/questions/2416706/total-spaces-of-universal-principal-g-bundles 
 A: There is a subtlety here that I think the classical literature doesn't deal with well.  Consider the following three statements for a principle $G$--bundle $E \rightarrow B$:
(a) $E$ is a terminal object in the homotopy category of (nice) free $G$--spaces.
(b) $E \rightarrow B$ is the universal $G$--bundle.
(c) $E$ is contractible.
Then (a) $\Leftrightarrow$ (b), and (c) $\Rightarrow$ (b), by basic bundle theory as in tom Dieck's book.  
It is also true that (b) $\Rightarrow$ (c), and the standard proof is as above: first one shows (e.g. using the Milnor construction) that there IS a universal bundle with contractible total space, and since universal bundles are unique, you are done.  But this proof has always seemed like cheating to me, so here is a direct proof that (a) $\Rightarrow$ (c):
Observe that $E$ satisfies (a) means that $[Y,E]_G$ is a singleton for all  free $G$--spaces $Y$. In particular, this is true when $Y=G \times X$, so we learn (since $[X,E] = [G \times X,E]_G$) that $[X,E]$ is a singleton for all spaces $X$. But this implies that $E$ is contractible.
