The right $J$-skew left unitor of $\mathbf{Rel}(A,B)$ is the natural transformation
\[ \lambda ^{\mathbf{Rel}(A,B),\rhd _{J}} \colon \mathbf{\lambda }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathbf{Rel}(A,B)} \Longrightarrow {\rhd _{J}}\circ {(\mathbb {1}^{\mathbf{Rel}(A,B)}_{\rhd }\times \mathsf{id})}, \]
as in the diagram
\[ \lambda ^{\mathbf{Rel}(A,B),\rhd _{J}}_{R} \colon R \hookrightarrow \underbrace{J\rhd _{J}R}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Ran}}_{J}(J)\mathbin {\diamond }R} \]
at $R$ is given by the composition
\begin{align*} R & \mkern 10mu\mathrlap {\mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}}\mkern 50mu \chi _{B}\mathbin {\diamond }R\\ & \mkern 10mu\mathrlap {\mathbin {\overset {\eta _{\chi _{B}}}{\Longrightarrow }}\mathbin {\diamond }\operatorname {\mathrm{id}}_{R}}\mkern 50mu \operatorname {\mathrm{Ran}}_{J}(J^{*}(\chi _{A}))\mathbin {\diamond }R\\ & \mkern 10mu\mathrlap {\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}}\mkern 50mu \operatorname {\mathrm{Ran}}_{J}(J^{*}\mathbin {\diamond }\chi _{A})\mathbin {\diamond }R\\ & \mkern 10mu\mathrlap {\mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}}\mkern 50mu \operatorname {\mathrm{Ran}}_{J}(J)\mathbin {\diamond }R\\ & \mkern 10mu\mathrlap {\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}}\mkern 50mu R\rhd _{J}J, \end{align*}
where $\eta \colon \operatorname {\mathrm{id}}_{\mathbf{Rel}(B,B)}\Longrightarrow \operatorname {\mathrm{Ran}}_{J}\circ J^{*}$ is the unit of the adjunction $J^{*}\dashv \operatorname {\mathrm{Ran}}_{J}$.
