Example 2.1.6.7 (00E4)

Example 2.1.6.7. Let $\operatorname{\mathcal{C}}$ be a monoidal category, and let $\ell : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ be the nonunital monoidal functor of Example 2.1.4.8 (carrying each object $X \in \operatorname{\mathcal{C}}$ to the functor $\ell _{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ given by $\ell _{X}(Y) = X \otimes Y$). Then $\ell $ is a monoidal functor: it admits a unit $\epsilon : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \ell _{ \mathbf{1} }$ given by the inverse of the left unit constraint of Construction 2.1.2.17. To prove this, it suffices to verify that $\epsilon $ satisfies property $(1)$ of Proposition 2.1.5.13 (Remark 2.1.6.4). Unwinding the definitions, this is equivalent to the assertion that for every object $X \in \operatorname{\mathcal{C}}$, the outer cycle of the diagram

\[ \xymatrix@C =50pt{ X \ar [dd]^{\operatorname{id}_ X} & \mathbf{1} \otimes X \ar [l]_-{ \lambda _ X} \ar [ddr]_{ \operatorname{id}_{ \mathbf{1} \otimes X} } & \mathbf{1} \otimes (\mathbf{1} \otimes X) \ar [l]_-{ \operatorname{id}_{ \mathbf{1} } \otimes \lambda _ X} \ar [d]^{ \alpha _{ \mathbf{1}, \mathbf{1}, X}} \\ & & (\mathbf{1} \otimes \mathbf{1}) \otimes X \ar [d]^{ \upsilon \otimes \operatorname{id}_ X} \\ X & & \mathbf{1} \otimes X \ar [ll]_{ \lambda _ X} } \]

is commutative. In fact, the whole diagram commutes: for the inner cycle on the left this is immediate, and for the inner cycle on the right it follows from the definition of the left unit constraing $\lambda _{X}$ (Construction 2.2.1.11).