# Kerodon

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Example 2.1.5.16. Let $\operatorname{\mathcal{C}}$ be a monoidal category, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor corepresented by the unit object $\mathbf{1} \in \operatorname{\mathcal{C}}$, given concretely by the formula $F(X ) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \mathbf{1}, X )$. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have a canonical map

$\mu _{X,Y}: F(X) \times F(Y) \rightarrow F( X \otimes Y ),$

which carries a pair of elements $x \in F(X)$, $y \in F(Y)$ to the composite map

$\mathbf{1} \xrightarrow { \upsilon ^{-1} } \mathbf{1} \otimes \mathbf{1} \xrightarrow {x \otimes y} X \otimes Y.$

The collection of maps $\{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ determines a lax monoidal structure on the functor $F$, with unit given by the map

$\epsilon : \{ \ast \} \rightarrow F(\mathbf{1}) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \mathbf{1}, \mathbf{1} ) \quad \quad \epsilon (\ast ) = \operatorname{id}_{ \mathbf{1} }.$