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 morphisms
$$
\mathcal C(B,C)\otimes\mathcal C(A,B)\to\mathcal C(A,C),
$$
and unit morphisms $I\to\mathcal C(A,A)$ satisfying the associative and unit diagrams. Its <underlying category of an enriched category>[underlying ordinary category] has hom-sets
$$
\mathcal V(I,\mathcal C(A,B)).
$$

If $\mathcal V$ is closed, take its internal hom $[A,B]$ as hom-object. Composition is the transpose of evaluation
$$
[B,C]\otimes[A,B]\otimes A\to[B,C]\otimes B\to C,
$$
and the unit is the transpose of $I\otimes A\cong A$. This is the <self-enrichment of a closed symmetric monoidal category>.

Solution

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

{wiki}

For a monoidal category $(\mathcal V,\otimes,I)$, a $\mathcal V$-enriched category has hom-objects in $\mathcal V$, composition morphisms $\mathcal C(B,C)\otimes\mathcal C(A,B)\to\mathcal C(A,C)$, and unit morphisms $I\to\mathcal C(A,A)$ satisfying associativity and unit laws.


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