Consider the diagram of finite CW complexes $X \stackrel{f}\leftarrow Y$ where $f$ is a cellular map and note that its *homotopy colimit* is precisely the mapping cylinder
$$C_H = \frac{X \sqcup (Y \times [0,1])}{\sim}$$
where $(y,0) \sim f(y)$. It is not too difficult to impose a nice CW structure on $C_H$ once you cellulate the unit interval: maybe the hardest thing here is to realize the product of finite CW complexes as a CW complex.

Here's my question:

Is there any hope whatsoever of imposing a (nice) CW structure on the homotopy limit of a dual diagram?

By dual diagram I mean a map $X \stackrel{g}\to Y$ of finite CW complexes where $g$ is any cellular map. Let $Y^I$ be the path space of all continuous $\gamma:[0,1] \to Y$, and note that the *homotopy limit* of our dual diagram is defined to be $$L_H = \{(x,\gamma) \in X \times Y^I\mid \gamma(0) = g(x)\}.$$ I don't see how to make the path space $Y^I$ a CW complex in any nice way, but maybe we can still do something with the subspace $L_H$.