Example 2.1.5.14 (00DP)

Example 2.1.5.14. In the special case where $\operatorname{\mathcal{C}}= \{ e \} $, we can identify a nonunital lax monoidal functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with a nonunital algebra object $A$ of $\operatorname{\mathcal{D}}$. In this case, Proposition 2.1.5.13 asserts that a morphism $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow A$ is a left unit (in the sense of Definition 2.1.5.1) if and only if the diagram

\[ \xymatrix@R =50pt@C=50pt{ \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{\epsilon \otimes \epsilon } \ar [d]^{\upsilon } & A \otimes A \ar [d]^{m} \\ \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{\epsilon } & A } \]

is commutative (that is, $\epsilon $ is idempotent) and the map

\[ \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes A \xrightarrow {\epsilon \otimes \operatorname{id}_{A} } A \otimes A \xrightarrow {m} A \]

is a monomorphism in $\operatorname{\mathcal{D}}$ (that is, $\epsilon $ is left cancellative). When $\operatorname{\mathcal{D}}$ is the category of sets (equipped with the cartesian monoidal structure of Example 2.1.3.2), this reduces to the statement of Proposition 2.1.2.3.