  Category is put to work in the non-associative realm in the article. We focus
on a typical example of non-associative category. Its objects are octonionic
bimodules, morphisms are octonionic para-linear maps, and compositions are
non-associative in general. The octonionic para-linear map is the main object
of octonionic Hilbert theory because of the octonionic Riesz representation
theorem. An octonionic para-linear map f is in general not octonionic linear
since it subjects to the rule Re (f (px)-pf (x))= 0. The composition should be
modified so that it preserves the octonionic para-linearity. In this
non-associative category, we introduce the Hom and Tensor functors which
constitute an adjoint pair. We establish the Yoneda lemma in terms of the new
notion of weak functor. To define the exactness in a non-associative category,
we introduce the notion of the enveloping category via a universal property.
This allows us to establish the exactness of the Hom functor and Tensor
functor.
