# Kerodon

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Exercise 2.1.7.9 (Uniqueness of Identities). Let $\operatorname{\mathcal{A}}$ be a monoidal category. A nonunital $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ consists of a collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$ of objects of $\operatorname{\mathcal{C}}$, together with objects $\{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \} _{X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})}$ of the category $\operatorname{\mathcal{A}}$ and composition laws

$c_{Z,Y,X}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$

which satisfy the associative law $(A)$ appearing in Definition 2.1.7.1. Show that, if a nonunital $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ can be promoted to an $\operatorname{\mathcal{A}}$-enriched category $\overline{\operatorname{\mathcal{C}}}$, then $\overline{\operatorname{\mathcal{C}}}$ is unique: that is, the identity maps $e_{X}: \mathbf{1} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ are determined by axiom $(U)$ of Definition 2.1.7.1.