Does this explain why the Borel construction is called the *homotopy orbit space*?

Does this explain why the Borel construction is called the *homotopy orbit
space*?

Let $X$ be a CW-complex and $G$ a topological group acting on $X$ by
$\sigma:G\times X\to X$.
Is the Borel construction $EG\times_G X$ weakly equivalent to
$\operatorname{hocolim}\left(G\times X
{\overset{\sigma}{\to}\atop\underset{pr}{\to}}X\right)$?
If not, is this at least true for a free action $\sigma$?
This would explain why the Borel construction is called the homotopy orbit
space.
The space $EG\times_G X$ can be defined as the orbit space $(EG\times
X)/G$ which is in turn defined by $$\operatorname{colim}\left(G\times
(EG\times X) {\overset{\tau}{\to}\atop\underset{pr}{\to}}EG\times
X\right)$$ and the action $\tau$ is the diagonal action on the factors
$EG$ and $X$.