The left $J$-skew monoidal product of $\mathbf{Rel}(A,B)$ is the functor
\[ \lhd _{J}\colon \mathbf{Rel}(A,B)\times \mathbf{Rel}(A,B) \to \mathbf{Rel}(A,B) \]
where
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Action on Objects. For each $R,S\in \operatorname {\mathrm{Obj}}(\mathbf{Rel}(A,B))$, we have
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Action on Morphisms. For each $R,S,R',S'\in \operatorname {\mathrm{Obj}}(\mathbf{Rel}(A,B))$, the action on $\operatorname {\mathrm{Hom}}$-sets
of $\lhd _{J}$ at $((R,S),(R',S'))$ is defined by1\[ (\lhd _{J})_{(G,F),(G',F')} \colon \operatorname {\mathrm{Hom}}_{\mathbf{Rel}(A,B)}(S,S')\times \operatorname {\mathrm{Hom}}_{\mathbf{Rel}(A,B)}(R,R') \to \operatorname {\mathrm{Hom}}_{\mathbf{Rel}(A,B)}(S\lhd _{J}R,S'\lhd _{J}R') \]for each $\beta \in \operatorname {\mathrm{Hom}}_{\mathbf{Rel}(A,B)}(S,S')$ and each $\alpha \in \operatorname {\mathrm{Hom}}_{\mathbf{Rel}(A,B)}(R,R')$.
- 1Since $\mathbf{Rel}(A,B)$ is posetal, this is to say that if $S\subset S'$ and $R\subset R'$, then $S\lhd _{J}R\subset S'\lhd _{J}R'$.
