I am looking at the proof of Brown's representability theorem (BRT) found at https://www.math.ru.nl/~gutierrez/files/Lecture13.pdf. Specifically, on pages 7-9, the author states and proves two lemmas that together suffice to prove BRT.
Let $F$ be a contravariant functor from the homotopy category of pointed connected CW complexes to the category of pointed sets that satisfies the wedge axiom and the Mayer-Vietoris axiom
We say that a pointed connected CW complex $B$ along with an element $b\in F(B)$ spherically represents $F$ if the set map $\nu_X : \left[X, B\right] \to F(X)$ given by $g\mapsto F(g)(b)$ is a natural bijection in $X\in \left\{S^n \mid n\in \mathbb{Z}_{\geq 1}\right\}$.
(Lemma 13.6) Let $X$ be a pointed CW complex and $x\in F(X)$. There exist an object $\left(B, b\right)$ spherically representing $F$ and a cofibration $f : X \to B$ such that $F(f)(b) =x$.
(Lemma 13.7) Any object spherically representing $F$ represents $F$.
The author uses 1 to prove 2. In his proof of 2, he invokes the existence of the map $f$ multiple times. But it seems to me that he never needs the fact that it's a cofibration. Am I missing something here?
No, there is no need for the map to be a cofibration. The second lemma is (or can be) proved entirely in the homotopy category, which knows nothing about cofibrations.