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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?

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I figured it out. The notation $[X,Y]$ in the article is for homotopy classes of maps, hence its enough to show it up to homotopy. $G$ sends weak equivalences to isomorphisms hence is equivalent to a functor out of the homotopy category, so the statement is in fact one about homotopy categories.

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    $\begingroup$ Just keep in mind that a model category is not there just to produce the homotopy category. When you do stuff in a model category you often get induced stuff on the homotopy category, but it is rarely the case that we are solely interested in the homotopy category. The model structure gives you a way to manage the localisation, but is interesting on its own. $\endgroup$ – Ittay Weiss Jan 5 '18 at 11:31

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