Definition 2.1.2.5. Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category. A unit of $\operatorname{\mathcal{C}}$ is a pair $( \mathbf{1}, \upsilon )$, where $\mathbf{1}$ is an object of $\operatorname{\mathcal{C}}$ and $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$ is an isomorphism, which satisfies the following additional condition:
- $(\ast )$
The functors
\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad C \mapsto \mathbf{1} \otimes C \]\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad C \mapsto C \otimes \mathbf{1} \]are fully faithful.