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. 



derived functor that preserves weak equivalences
The answer is yes.


Consider the functor $F:A \to Ho(B)$ obtained by composing $F:A \to B$ with the localization functor $B \to Ho(B)$.


Then this functor takes weak equivalences in $A$ to isomorphisms in $Ho(B)$,
so by the [universal property of localization](https://ncatlab.org/nlab/show/localization#definition), there is a functor $F':Ho(A) \to Ho(B)$, such that $F \cong F'\circ Q$, where $Q:A \to Ho(A)$ is the localization functor. This functor $F'$ is your derived functor.


