In this lecture I introduce two possible notions of what it might mean for a functor with domain the homotopy category of a model category M to be “induced” by a functor out of M itself.  These two notions, called left and right derived functors, are inspired by the notion of derived functor in homological algebra.  

Using the description of the homotopy category as a localization, I prove that any functor with domain a model category that sends all weak equivalences to isomorphisms induces both left and right derived functors, which are in fact the same.