The right $J$-skew associator of $\mathbf{Rel}(A,B)$ is the natural transformation
\[ \alpha ^{\mathbf{Rel}(A,B),\rhd _{J}}\colon {\rhd _{J}}\circ {(\mathsf{id}\times {\rhd _{J}})}\Longrightarrow {\rhd _{J}}\circ {({\rhd _{J}}\times \mathsf{id})}\circ {\mathbf{\alpha }^{\mathsf{Cats},-1}_{\mathbf{Rel}(A,B),\mathbf{Rel}(A,B),\mathbf{Rel}(A,B)}}, \]
\[ \alpha ^{\mathbf{Rel}(A,B),\rhd _{J}}_{T,S,R}\colon \underbrace{T\rhd _{J}(S\rhd _{J}R)}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Ran}}_{J}(T)\mathbin {\diamond }\operatorname {\mathrm{Ran}}_{J}(S)\mathbin {\diamond }R}\hookrightarrow \underbrace{(T\rhd _{J}S)\rhd _{J}R}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Ran}}_{J}(\operatorname {\mathrm{Ran}}_{J}(T)\mathbin {\diamond }S)\mathbin {\diamond }R} \]
at $(T,S,R)$ is given by
\[ \alpha ^{\mathbf{Rel}(A,B),\rhd }_{T,S,R}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\gamma \mathbin {\diamond }\operatorname {\mathrm{id}}_{R}, \]
where
\[ \gamma \colon \operatorname {\mathrm{Ran}}_{J}(T)\mathbin {\diamond }\operatorname {\mathrm{Ran}}_{J}(S) \hookrightarrow \operatorname {\mathrm{Ran}}_{J}(\operatorname {\mathrm{Ran}}_{J}(T)\mathbin {\diamond }S) \]
is the inclusion adjunct to the inclusion
\[ \operatorname {\mathrm{id}}_{\operatorname {\mathrm{Ran}}_{J}(T)}\mathbin {\diamond }\epsilon _{S}\colon \underbrace{\operatorname {\mathrm{Ran}}_{J}(T)\mathbin {\diamond }\operatorname {\mathrm{Ran}}_{J}(S)\mathbin {\diamond }J}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}J^{*}(\operatorname {\mathrm{Ran}}_{J}(T)\mathbin {\diamond }\operatorname {\mathrm{Ran}}_{J}(S))}\hookrightarrow \operatorname {\mathrm{Ran}}_{J}(T)\mathbin {\diamond }S \]
under the adjunction $J^{*}\dashv \operatorname {\mathrm{Ran}}_{J}$, where $\epsilon \colon {\operatorname {\mathrm{Ran}}_{J}}\mathbin {\diamond }{J}\Longrightarrow \operatorname {\mathrm{id}}_{\mathbf{Rel}(A,B)}$ is the counit of the adjunction $J^{*}\dashv \operatorname {\mathrm{Ran}}_{J}$.
