# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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.