Yeah, for example in chapter 0 he proves that the quotient map for a pair satisfying the homotopy extension property with the subspace being contractible, the quotient map is a homotopy equivalence. He does so by taking a homotopy $f_t$ extending a homotopy equivalence from the subspace to a point and then factors each $f_t$ individually to a quotient map $\tilde{f}_t$.
These $\tilde{f}_t$ are supposed to be the final homotopy, but he doesn't justify why they are still jointly continuous in both variables (this comes down to the fact that the product of a quotient map with the identity map …