  This paper develops a general theory of canonical bases, and how they arise
naturally in the context of categorification. As an application, we show that
Lusztig's canonical basis in the whole quantized universal enveloping algebra
is given by the classes of the indecomposable 1-morphisms in a categorification
when the associated Lie algebra is finite type and simply laced. We also
introduce natural categories whose Grothendieck groups correspond to the tensor
products of lowest and highest weight integrable representations. This
generalizes past work of the author's in the highest weight case.
