You can show the evaluation map is a weak fibration, which is good enough for many purposes, including the formation of homotopy pullbacks.

You can show the evaluation map is a weak fibration (I think this is the term: I mean a map homotopy equivalent in the category of spaces over $X\times X$ to a fibration -- namely $X^I\to X\times X$), which is good enough for many purposes, including the formation of homotopy pullbacks.