Definition Locally graded category [efr-002I]

Let \mathcal {M} be a monoidal category. Recall that we may equip the presheaf category [\mathcal {M}^\mathrm {op}, \mathsf {Set}] with the monoidal structure of Day convolution. A locally graded \mathcal {M}-category is a category enriched in [\mathcal {M}^\mathrm {op}, \mathsf {Set}]

Equivalently, a locally graded \mathcal {M}-category consists of

1. A collection of objects
2. For each pair of objects X,Y and M \in \mathcal {M}, a collection of M-morphisms \mathcal {C}_M(X,Y)
3. For each morphism M' \to M, a function \mathcal {C}_{M'}(X,Y) \to \mathcal {C}_M(X,Y)
4. A composition operation \mathcal {C}_{M}(X,Y) \times \mathcal {C}_{M'}(Y,Z) \to \mathcal {C}_{M \otimes M'}(X,Z)
5. For each object X, an identity morphism in \mathcal {C}_{I_\mathcal {M}}(X,X)
6. Satisfying evident laws