such that $R$ is monic: for every object $A$ in $\mathcal{C}$ and morphisms $f: A \to E$ and $g: A \to E$, $R \circ f = R \circ g$ implies that $f = g$.

In general categories

A loop digraph object in a category $\mathcal{C}$ is

such that $s$ and $t$ are jointly monic: for every object $A$ in $\mathcal{C}$ and morphisms $f: A \to E$ and $g: A \to E$, $s \circ f = s \circ g$ and $t \circ f = t \circ g$ imply that $f = g$.