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?