Definition 6.4.1.1.1 (00MQ): The Left $J$-Skew Monoidal Product of $\mathbf{Rel}\webleft (A,B\webright )$

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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 $\alpha \in \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (R,R'\webright )$.

¹Since $\mathbf{Rel}\webleft (A,B\webright )$ is posetal, this is to say that if $S\subset S'$ and $R\subset R'$, then $S\lhd _{J}R\subset S'\lhd _{J}R'$.