3.3.3 The Left Annihilator

The left annihilator of the product of sets is the natural isomorphism

\[ \zeta ^{\mathsf{Sets}}_{\ell } \colon \mathbb {0}^{\mathsf{Sets}}\circ \mathbf{\mu }^{\mathsf{Cats}_{\mathsf{2}}}_{4|\mathsf{Sets},\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}\circ \webleft (\text{id}_{\mathsf{pt}}\times \mathbf{\epsilon }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}}\webright ) \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\times \circ \webleft (\mathbb {0}^{\mathsf{Sets}}\times \text{id}_{\mathsf{Sets}}\webright ) \]

as in the diagram

with components

\[ \zeta ^{\mathsf{Sets}}_{\ell |A}\colon \text{Ø}\times A\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\text{Ø}. \]


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