  We distinguish between faint, weak, strong and strict localizations of
categories at morphism families and show that this framework captures the
different types of derived functors that are considered in the literature. More
precisely, we show that Kan and faint derived functors coincide when we use the
classical Kan homotopy category, and when we use the Quillen homotopy category,
Kan and strong derived functors coincide. Our comparison results are based on
the fact that the Kan homotopy category is a weak localization and that the
Quillen homotopy category is a strict localization.
