The category $\mathbf{Rel}(A,B)$ admits a right 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 Right Skew Monoidal Product. The right $J$-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 Right 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 Right Skew Associators. The natural transformation
of Definition 8.9.3.1.1.\[ \alpha ^{\mathbf{Rel}(A,B),\rhd _{J}}\colon {\rhd _{J}}\circ {(\mathsf{id}\times {\rhd _{J}})}\Longrightarrow {\rhd _{J}}\circ {({\rhd _{J}}\times \mathsf{id})}\circ {\mathbf{\alpha }^{\mathsf{Cats},-1}_{\mathbf{Rel}(A,B),\mathbf{Rel}(A,B),\mathbf{Rel}(A,B)}} \] -
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The Right Skew Left Unitors. The natural transformation
\[ \lambda ^{\mathbf{Rel}(A,B),\rhd _{J}} \colon \mathbf{\lambda }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathbf{Rel}(A,B)} \Longrightarrow {\rhd _{J}}\circ {(\mathbb {1}^{\mathbf{Rel}(A,B)}_{\rhd }\times \mathsf{id})} \] -
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The Right Skew Right Unitors. The natural transformation
\[ \rho ^{\mathbf{Rel}(A,B),\rhd _{J}} \colon {\rhd _{J}}\circ {(\mathsf{id}\times \mathbb {1}^{\mathbf{Rel}(A,B)}_{\rhd })} \Longrightarrow \mathbf{\rho }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathbf{Rel}(A,B)} \]
