Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 5 / a / Solution

Let $(\mathcal{V},\otimes ,I)$ be a symmetric monoidal category. A $\mathcal{V}$-enriched category has objects, hom-objects $\mathcal{C}(A,B)\in \mathcal{V}$, composition morphismsand unit morphisms $I\to \mathcal{C}(A,A)$ satisfying the associative and unit diagrams. Its underlying ordinary category has hom-sets

If $\mathcal{V}$ is closed, take its internal hom $[A,B]$ as hom-object. Composition is the transpose of evaluationand the unit is the transpose of $I\otimes A\cong A$. This is the self-enrichment of a closed symmetric monoidal category.