The category $\mathbf{Rel}(A,B)$ admits a left skew monoidal category structure consisting of
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The Underlying Category. The posetal category associated to the poset $\mathbf{Rel}(A,B)$ of relations from $A$ to $B$ of
of .
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The Left Skew Monoidal Product. The left $J$-skew monoidal product
\[ \lhd _{J}\colon \mathbf{Rel}(A,B)\times \mathbf{Rel}(A,B)\to \mathbf{Rel}(A,B) \] -
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The Left Skew Monoidal Unit. The functor
\[ \mathbb {1}^{\mathbf{Rel}(A,B),\lhd _{J}} \colon \mathsf{pt}\to \mathbf{Rel}(A,B) \] -
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The Left Skew Associators. The natural transformation
of Definition 8.8.3.1.1.\[ \alpha ^{\mathbf{Rel}(A,B),\lhd _{J}}\colon {\lhd _{J}}\circ {({\lhd _{J}}\times \mathsf{id})}\Longrightarrow {\lhd _{J}}\circ {(\mathsf{id}\times {\lhd _{J}})}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathbf{Rel}(A,B),\mathbf{Rel}(A,B),\mathbf{Rel}(A,B)}} \] -
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The Left Skew Left Unitors. The natural transformation
\[ \lambda ^{\mathbf{Rel}(A,B),\lhd _{J}} \colon {\lhd _{J}}\circ {({\mathbb {1}^{\mathbf{Rel}(A,B)}_{\lhd _{J}}}\times {\mathsf{id}})} \Longrightarrow \mathbf{\lambda }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathbf{Rel}(A,B)} \] -
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The Left Skew Right Unitors. The natural transformation
\[ \rho ^{\mathbf{Rel}(A,B),\lhd _{J}} \colon \mathbf{\rho }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathbf{Rel}(A,B)} \Longrightarrow {\lhd _{J}}\circ {(\mathsf{id}\times \mathbb {1}^{\mathbf{Rel}(A,B)}_{\lhd _{J}})} \]
