How to use Ken Brown's lemma in page 11 of "Model Categories"?

Question

I was reading Mark Hovey's "Model Categories".

In the proof of the proposition 1.2.5. (iv) (page 11), it says we can see that when $B$ is cofibrant and $h:X\to Y$ is a weak equivalence of fibrant objects, $h$ induces an isomorphism $$\mathcal{C}(B, X)/\overset{\ell}\sim\xrightarrow{\cong}\mathcal{C}(B, Y)/\overset{\ell}\sim $$ using Ken Brown's lemma and the case in which $h$ is a trivial fibration.

It seems that this is a basic argument (similar one is found in the first paraghraph of the proof of the proposition 1.2.8). How can I use Ken Brown's lemma in these proofs?

Answer 1

K.B. says that any functor $F:M\to C$ from a model category to a category with a class of weak equivalences satisfying 2-for-3 which sends trivial (co)fibrations between (co)fibrant objects to weak equivalences must send all weak equivalences between (co)fibrant objects to weak equivalences. We have the functor $C(B,-)$, which we already know sends trivial fibrations to "weak equivalences of sets", that is, isomorphisms, after modding out left homotopy. The result follows.

I think Hovey may only state K.B. for model categories as codomain as well, but the proof won't use that. Anyway, you can take a trivial model structure on sets in this case, if you want.