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