Let $ \mathcal{M} $ be a model category and $ A $ a object of $ \mathcal{M} $. The "choice category" $ \mathcal{C} $ of cofibrant approximations of $ A $ is the full subcategory of $ \mathcal{M}_{/A} $ such that $ X \to A \in \operatorname{Ob} \mathcal{C} $ if and only if $ X \to A $ is a weak equivalence and $ X $ is cofibrant.
How can I prove that (the nerve of) $ \mathcal{C} $ is contractible?
Hirschhorn's book "Model categories and their localizations" has a more general fact as Theorem 14.5.4, but its proof seems to be incorrect. (In his proof, $ (F\mathbf{X}, p_{\mathbf{X}}) $ can be not terminal.)
Edit: I require the factorizations are functorial as in the book of Hirschhorn.
Consider the full subcategory W(M/A)⊂M/A consisting of objects that are weak equivalences in M. The latter category in its turn has a full subcategory C(W(M/A)) consisting of objects with a cofibrant source. The inclusion functor C(W(M/A))→W(M/A) and the cofibrant replacement functor W(M/A)→C(W(M/A)) (induced from the cofibrant replacement functor on the model category M/A) induce an equivalence on nerves: the composition K: W(M/A)→C(W(M/A))→W(M/A) admits a natural weak equivalence K→id and the composition L: C(W(M/A))→W(M/A)→C(W(M/A)) admits a natural weak equivalence L→id. Thus the nerve of C(W(M/A)) is equivalent to the nerve of W(M/A), and the latter is contractible because it admits a natural weak equivalence id→const_1, where 1 is the terminal object.