The left $J$-skew associator of $\mathbf{Rel}\webleft (A,B\webright )$ is the natural transformation
\[ \alpha ^{\mathbf{Rel}\webleft (A,B\webright ),\lhd _{J}}\colon {\lhd _{J}}\circ {\webleft ({\lhd _{J}}\times \mathsf{id}\webright )}\Longrightarrow {\lhd _{J}}\circ {\webleft (\mathsf{id}\times {\lhd _{J}}\webright )}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathbf{Rel}\webleft (A,B\webright ),\mathbf{Rel}\webleft (A,B\webright ),\mathbf{Rel}\webleft (A,B\webright )}}, \]
whose component
\[ \alpha ^{\mathbf{Rel}\webleft (A,B\webright ),\lhd _{J}}_{T,S,R}\colon \underbrace{\webleft (T\lhd _{J}S\webright )\lhd _{J}R}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}T\mathbin {\diamond }\text{Rift}_{J}\webleft (S\webright )\mathbin {\diamond }\text{Rift}_{J}\webleft (R\webright )}\hookrightarrow \underbrace{T\lhd _{J}\webleft (S\lhd _{J}R\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}T\mathbin {\diamond }\text{Rift}_{J}\webleft (S\mathbin {\diamond }\text{Rift}_{J}\webleft (R\webright )\webright )} \]
at $\webleft (T,S,R\webright )$ is given by
\[ \alpha ^{\mathbf{Rel}\webleft (A,B\webright ),\lhd _{J}}_{T,S,R}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{T}\mathbin {\diamond }\gamma , \]
where
\[ \gamma \colon \text{Rift}_{J}\webleft (S\webright )\mathbin {\diamond }\text{Rift}_{J}\webleft (R\webright ) \hookrightarrow \text{Rift}_{J}\webleft (S\mathbin {\diamond }\text{Rift}_{J}\webleft (R\webright )\webright ) \]
is the inclusion adjunct to the inclusion
\[ \epsilon _{S}\mathbin {\star }\text{id}_{\text{Rift}_{J}\webleft (R\webright )} \colon \underbrace{J\mathbin {\diamond }\text{Rift}_{J}\webleft (S\webright )\mathbin {\diamond }\text{Rift}_{J}\webleft (R\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}J_{*}\webleft (\text{Rift}_{J}\webleft (S\webright )\mathbin {\diamond }\text{Rift}_{J}\webleft (R\webright )\webright )} \hookrightarrow S\mathbin {\diamond }\text{Rift}_{J}\webleft (R\webright ) \]
under the adjunction $J_{*}\dashv \text{Rift}_{J}$, where $\epsilon \colon {J\mathbin {\diamond }\text{Rift}_{J}}\Longrightarrow \text{id}_{\mathbf{Rel}\webleft (A,B\webright )}$ is the counit of the adjunction $J_{*}\dashv \text{Rift}_{J}$.
