Concretely, $\smash {\mathord {\sim }^{\mathrm{trans}}_{R}}$ is the free non-unital monoid on $R$ in $\webleft (\mathbf{Rel}\webleft (A,A\webright ),\mathbin {\diamond }\webright )$, being given by
\begin{align*} R^{\mathrm{trans}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\coprod _{n=1}^{\infty }R^{\mathbin {\diamond }n}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcup _{n=1}^{\infty }R^{\mathbin {\diamond }n}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (a,b\webright )\in A\times B\ \middle |\ \begin{aligned} & \text{there exists some $\webleft (x_{1},\ldots ,x_{n}\webright )\in R^{\times n}$}\\ & \text{such that $a\sim _{R}x_{1}\sim _{R}\cdots \sim _{R}x_{n}\sim _{R}b$}\end{aligned} \webright\} .\end{align*}