Enriched category

Definition

Let $(\mathcal{V}, \otimes, I, \alpha, \lambda, \rho)$ be a monoidal category. A $\mathcal{V}$-enriched category (or $\mathcal{V}$-category) $\mathcal{C}$ consists of:

1. a class $\mathrm{Ob}(\mathcal{C})$ of objects;
2. for each pair of objects $a, b$, an object $\mathcal{C}(a, b)$ of $\mathcal{V}$, the hom-object;
3. for each triple $a, b, c$, a morphism $M_{a,b,c} \colon \mathcal{C}(b, c) \otimes \mathcal{C}(a, b) \to \mathcal{C}(a, c)$ of $\mathcal{V}$, the composition;
4. for each object $a$, a morphism $j_a \colon I \to \mathcal{C}(a, a)$ of $\mathcal{V}$, the identity.

These data satisfy, for all objects $a, b, c, d$:

1. associativity: $M_{a,b,d} \circ (M_{b,c,d} \otimes \id) = M_{a,c,d} \circ (\id \otimes M_{a,b,c}) \circ \alpha$ as morphisms $(\mathcal{C}(c, d) \otimes \mathcal{C}(b, c)) \otimes \mathcal{C}(a, b) \to \mathcal{C}(a, d)$;
2. units: $M_{a,b,b} \circ (j_b \otimes \id) = \lambda$ and $M_{a,a,b} \circ (\id \otimes j_a) = \rho$ as morphisms $I \otimes \mathcal{C}(a, b) \to \mathcal{C}(a, b)$ and $\mathcal{C}(a, b) \otimes I \to \mathcal{C}(a, b)$.