Homotopy limits commute I'm looking for a reference for the fact that homotopy limits commute. That is 
$\mathrm{holim}_U \mathrm{holim}_W E = \mathrm{holim}_W\mathrm{holim}_U E $.
Preferably looking for a result that define homotopy limit as the right derived functor of $\lim$. 
 A: I couldn't find a reference for this either, so came up with this argument of the cuff. 
We can define homotopy limits as the right derived functor of $\lim$, and recall that this is a right adjoint - this is really the crucial thing to note.
Then $\mathrm{holim}_U \mathrm{holim}_W E = \mathrm{holim}_U \lim_W RE$, where $R$ denotes fibrant replacement. 
(Note that I am implicitly assuming that $E$ is in a nice model category with a functorial factorisation, so the fibrant replacement functor exists)
Now $\mathrm{holim}$ is a right adjoint functor hence commutes with limits, it follows that 
$\mathrm{holim}_U \lim_W RE = \lim_W \mathrm{holim}_U RE$
Now since $RE$ is fibrant already, this last is, 
 $$ \lim_W \lim_U RE$$.
Now if you take the other side of the equation, and follow the same process we get
$$ \mathrm{holim}_W \mathrm{holim}_U E = \mathrm{holim}_W \lim_U RE = \lim_U \mathrm{holim}_W RE = \lim_U \lim_W RE = \lim_W \lim_U RE$$
with the last equality following by the fact limits commute [ this is just a special case of right adjoints preserving limits]
