Kerodon

Remark 2.1.7.6. Let $\operatorname{\mathcal{A}}$ be a monoidal category and let $\operatorname{\mathcal{C}}$ be an ordinary category. We define an $\operatorname{\mathcal{A}}$-enrichment of $\operatorname{\mathcal{C}}$ to be an $\operatorname{\mathcal{A}}$-enriched category $\widetilde{\operatorname{\mathcal{C}}}$ together with an identification of $\operatorname{\mathcal{C}}$ with the underlying category of $\widetilde{\operatorname{\mathcal{C}}}$, in the sense of Example 2.1.7.5.