Proposition [efr-002J]

Recall that the Yoneda embedding \mathcal {M} \hookrightarrow [\mathcal {M}^\mathrm {op},\mathsf {Set}] is strong monoidal. Hence it induces a functor from \mathcal {M}-categories to locally \mathcal {M}-graded categories, which carries \mathcal {C} to a locally graded category with \mathcal {C}_M(X,Y) = \mathcal {M}(M, \mathcal {C}(X,Y))