Categories graded by group homomorphisms

We generalise to a group homomorphism $\tau$ the $\chi$-graded categories of Sözer and Virelizier. These are categories in which both morphisms and objects have compatible degrees. We give a ‘half-enriched’ Yoneda lemma, a structure theorem for semisimple $\tau$-graded categories, and an alternative picture of $\tau$-graded categories in terms of pseudofunctors into $\mathbf{C}\mathbf{a}\mathbf{t}$.