6.4.1 The Left Skew Monoidal Product
Let $A$ and $B$ be sets and let $J\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.
The left $J$-skew monoidal product of $\mathbf{Rel}\webleft (A,B\webright )$ is the functor
\[ \lhd _{J}\colon \mathbf{Rel}\webleft (A,B\webright )\times \mathbf{Rel}\webleft (A,B\webright ) \to \mathbf{Rel}\webleft (A,B\webright ) \]
where
- Action on Objects. For each $R,S\in \text{Obj}\webleft (\mathbf{Rel}\webleft (A,B\webright )\webright )$, we have
- Action on Morphisms. For each $R,S,R',S'\in \text{Obj}\webleft (\mathbf{Rel}\webleft (A,B\webright )\webright )$, the action on $\textup{Hom}$-sets
\[ \webleft (\lhd _{J}\webright )_{\webleft (G,F\webright ),\webleft (G',F'\webright )} \colon \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (S,S'\webright )\times \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (R,R'\webright ) \to \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (S\lhd _{J}R,S'\lhd _{J}R'\webright ) \]
of $\lhd _{J}$ at $\webleft (\webleft (R,S\webright ),\webleft (R',S'\webright )\webright )$ is defined by1 for each $\beta \in \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (S,S'\webright )$ and each $\alpha \in \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (R,R'\webright )$.