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Example (Enrichment in Vector Spaces). Let $k$ be a field and let $\operatorname{Vect}_{k}$ denote the category of vector spaces over $k$, endowed with the monoidal structure given by tensor product over $k$ (Example Then choosing an $\operatorname{Vect}_{k}$-enrichment of $\operatorname{\mathcal{C}}$ is equivalent to endowing each of the sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ with the structure of a $k$-vector space, for which the composition maps

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z) \]

are $k$-bilinear.