related by the Dold-Kan correspondence
Let be a category of fibrant objects.
Every morphism in factors as
Take to be the composite vertical morphism here:
To see that this is indeed a fibration, notice that, by the pasting law, the above pullback diagram can be refined to a double pullback diagram as follows
Both squares are pullback squares. Since pullbacks of fibrations are fibrations, the morphism is a fibration. Similarly, since is assumed to be fibrant, also the projection map is a fibration.
Since is therefore the composite
of two fibrations, it is itself a fibration.
That establishes the claim.
Let be a functor from a category of fibrant objects to any category with weak equivalences that sends trivial fibrations to weak equivalences. Then this functor necessarily sends all weak equivalences to weak equivalences, hence is a homotopical functor.
If is a weak equivalence, then by 2-out-of-3 also the from the above proof is a weak equivalence, hence a trivial fibration.
Apply the functor to the diagram of the above proof
By the assumption that preserves trivial fibrations we have that both horizontal morphisms as well as the total vertical morphism and the bottom vertical morphism are weak equivalences. By 2-out-of-3 it then follows that also the top left vertical morphism is a weak equivalence and then finally that is.
If is the full subcategory of fibrant objects in a model category, then this corollary asserts that a right Quillen functor , which by its axioms is required only to preserve fibrations and trivial fibrations, preserves also weak equivalences between fibrant objects.
presents the homotopy pullback of the original diagram.
For instance page 4 of