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It's a weak equivalence by the fact that it's a right proper model category (since all objects are fibrant), and therefore the fiber of a fibration is a homotopy pullback (and a hofiber). In particular, the induced map between two representatives of a homotopy pullback is necessarily a weak equivalence.

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It's a weak equivalence by the fact that it's a right proper model category (since all objects are fibrant), and therefore the fiber of a fibration is a homotopy pullback.