tensor unit

Let \(\mathcal{C}\) be a monoidal category. Then the tensor unit \(1 \in \mathcal{C}\) is an object such that for the tensor functor it holds for the identity functor \(\mathrm{id}_{\mathcal{C}}\)

\begin{align*} \otimes(1, -) =& \mathrm{id}_{\mathcal{C}} \\ \otimes(-, 1) =& \mathrm{id}_{\mathcal{C}} \end{align*}