It seems like homotopy pullback is pullback for the case of D(CW) (derived category of CW-complexes). Is this true?
In general, I want to say that homotopy pullback satisfies the universal property of pullback in a derived category.
What I know for sure is that the pullback of fibrations is homotopy pullback. In a derived category such as D(CW/X), weak equivalences are isomorphisms, and each map can be replaced by a fibration. Hence, up to isomorphism, everything is a fibration, on which the homotopy pullback is the pullback.
I wanted someone to confirm that this homotopy pullback (which coincides with the pullback after replacement) satisfies the universal property of pullback.