In Jardine's article on the subject: https://ncatlab.org/nlab/files/JardineBrownrep.pdf
He shows a version of Browns representability, which asserts the representability of a functor out of a model category (Theorem 24), as opposed to the classical result which is out of the homotopy category of CW-complexes.
Why is this not a statement about functors out of the homotopy category associated to the model category?
As stated just above Lemma 22 in the article the proofs are the same as for some of the lemmas earlier in the text. I have tried to reproduce the proof of theorem 24, by using the strategy of Proposition 12, but i ran into some problems which i think relate to the above question. The surjectivity is easy, but for injectivity i replace $X \otimes \Delta^1$ with the path object $X^I$ to obtain a homotopy the same way he does, but then $u^*$ ends up being only injective upto (right) homotopy.
Is this somehow enough?