Conjugate pairs of categories and Quillen equivalent stable model categories of functors

We describe the notion of a conjugate pair (B,A) of small categories, wherein maps in B admit a factorization by maps in the subcategory A, much in the spirit of a “two-sided” calculus of fractions. When (B,A) is a conjugate pair, we prove that for any cofibrantly generated model category C there is an induced Quillen adjunction between the functor categories [Bop,C] and [Aop,C]. When C is a left proper stable model category, this adjunction is a Quillen equivalence. Finally, we demonstrate that minor modifications of our arguments give the analogous result when C is instead assumed to be an Ab-category.