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Construction 2.1.2.17 (Left and Right Unit Constraints). Let $\operatorname{\mathcal{C}}= (\operatorname{\mathcal{C}}, \otimes , \alpha , \mathbf{1}, \upsilon )$ be a monoidal category. For each object $X \in \operatorname{\mathcal{C}}$, we have canonical isomorphisms

\[ \mathbf{1} \otimes ( \mathbf{1} \otimes X) \xrightarrow { \alpha _{ \mathbf{1}, \mathbf{1}, X} } ( \mathbf{1} \otimes \mathbf{1} ) \otimes X \xrightarrow { \upsilon \otimes \operatorname{id}_ X} \mathbf{1} \otimes X. \]

Since the functor $Y \mapsto \mathbf{1} \otimes Y$ is fully faithful, it follows that there is a unique isomorphism $\lambda _{X}: \mathbf{1} \otimes X \xrightarrow {\sim } X$ for which the diagram

\[ \xymatrix { \mathbf{1} \otimes (\mathbf{1} \otimes X) \ar [rr]^-{\alpha _{\mathbf{1}, \mathbf{1}, X} }_{\sim } \ar [dr]_{ \operatorname{id}_\mathbf {1} \otimes \lambda _ X}^{\sim } & & (\mathbf{1} \otimes \mathbf{1}) \otimes X \ar [dl]^{ \upsilon \otimes \operatorname{id}_ X}_{\sim } \\ & \mathbf{1} \otimes X & } \]

commutes. We will refer to $\lambda _{X}$ as the left unit constraint. Similarly, there is a unique isomorphism $\rho _{X}: X \otimes \mathbf{1} \simeq X$ for which the diagram

\[ \xymatrix { X \otimes (\mathbf{1} \otimes \mathbf{1}) \ar [rr]^-{\alpha _{X, \mathbf{1}, \mathbf{1}} }_{\sim } \ar [dr]_{ \operatorname{id}_ X \otimes \upsilon }^{\sim } & & (X \otimes \mathbf{1}) \otimes \mathbf{1} \ar [dl]^{ \rho _ X \otimes \operatorname{id}_{ \mathbf{1} } }_{\sim } \\ & X \otimes \mathbf{1} & } \]

commutes; we refer to $\rho _ X$ as the right unit constraint.