The Clowder Project

The left $J$-skew monoidal product of $\mathbf{Rel}\webleft (A,B\webright )$ is the functor

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 by^[1]
  
  for each $\beta \in \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (S,S'\webright )$ and each $\alpha \in \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (R,R'\webright )$.