In the paper http://arxiv.org/pdf/math/0101162.pdf, the authors claim during the proof of Prop. 4.2 that a functor $F:A \to B$ which preserves fibrations and weak equivalences preserves homotopy cartesian squares. It seems to me that without asking it to commute with ordinary pullbacks the claim is false.

Moreover, it is also stated that the induced functor $A^C \to B^C$, where $C$ is a Reedy category, preserves Reedy fibration. Again, I think it has to commute with pullbacks (e.g. a right Quillen would satisfy everything of the above for fibrant objects).

Am I right or am I missing something? Thanks in advance for any help.