Suppose we have a functor $F:A\rightarrow B$ between model categories.
1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived functor: $$Ho(F): Ho(A)\rightarrow Ho(B) $$ 2- Assume that F takes weak equivalences to weak equivalences, can we define the derived functor: $$Ho(F): Ho(A)\rightarrow Ho(B) $$
It will be great to see the construction of such derived functor in detail. Thank you.