An elegant Reedy category is a Reedy category$R$ such that the following equivalent conditions hold

For every monomorphism$A\hookrightarrow B$ of presheaves on $R$ and every $x\in R$, the induced map $A_x \amalg_{L_x A} L_x B \to B_x$ is a monomorphism.

Joyal’s disk categories$\Theta_n$ are elegant Reedy categories.

Every direct category is a Reedy category with no degeneracies, hence trivially an elegant one.

If $X$ is any presheaf on an elegant Reedy category $R$, then the opposite of its category of elements$(el X)^{op}$ is again an elegant Reedy category. This is fairly easy to see from the fact that $Set^{el X}$ is equivalent to the slice category $Set^{R^{op}}/X$.

Note that unlike the notion of Reedy category, the notion of elegant Reedy category is not self-dual: if $R$ is elegant then $R^{op}$ will not generally be elegant.