Definition 2.1.5.1. Let $\operatorname{\mathcal{C}}$ be a monoidal category with unit object $\mathbf{1}$, and let $A$ be a nonunital algebra object of $\operatorname{\mathcal{C}}$ (Example 2.1.4.11) with multiplication $m: A \otimes A \rightarrow A$. We say that a morphism $\epsilon : \mathbf{1} \rightarrow A$ is a left unit for $A$ if the composite map
\[ A \xrightarrow { \lambda _{A}^{-1} } \mathbf{1} \otimes A \xrightarrow { \epsilon \otimes \operatorname{id}_ A} A \otimes A \xrightarrow {m} A \]
is the identity map from $A$ to itself; here $\lambda _{A}: \mathbf{1} \otimes A \xrightarrow {\sim } A$ denotes the left unit constraint of Construction 2.1.2.17. We say that $\epsilon $ is a right unit of $A$ if the composite map
\[ A \xrightarrow { \rho _{A}^{-1} } A \otimes \mathbf{1} \xrightarrow { \operatorname{id}_{A} \otimes \epsilon } A \otimes A \xrightarrow {m} A \]
is equal to the identity. We say that $\epsilon $ is a unit of $A$ if it is both a left and a right unit of $A$.