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