  We introduce a new categorical framework for studying derived functors, and
in particular for comparing composites of left and right derived functors. Our
central observation is that model categories are the objects of a double
category whose vertical and horizontal arrows are left and right Quillen
functors, respectively, and that passage to derived functors is functorial at
the level of this double category. The theory of conjunctions and mates in
double categories, which generalizes the theory of adjunctions and mates in
2-categories, then gives us canonical ways to compare composites of left and
right derived functors. We give a number of sample applications, most of which
are improvements of existing proofs in the literature.
