  We classify all fusion categories for a given set of fusion rules with three
simple object types. If a conjecture of Ostrik is true, our classification
completes the classification of fusion categories with three simple object
types. To facilitate the discussion we describe a convenient, concrete and
useful variation of graphical calculus for fusion categories, discuss
pivotality and sphericity in this framework, and give a short and elementary
re-proof of the fact that the quadruple dual functor is naturally isomorphic to
the identity.
