  Given an appropriate diagram of left Quillen functors between model
categories, one can define a notion of homotopy fiber product, but one might
ask if it is really the correct one. Here, we show that this homotopy pullback
is well-behaved with respect to translating it into the setting of more general
homotopy theories, given by complete Segal spaces, where we have well-defined
homotopy pullbacks.
