A loop graph object in a category $\mathcal{C}$ with finite products is a loop digraph object$(V, E, R:E \to V \times V)$ with a morphism $i:E \to E$ such that

$p_1 \circ R = p_2 \circ R \circ sym$

$p_2 \circ R = p_1 \circ R \circ sym$

$i \circ i = id_E$

where $p_1, p_2:V \times V \to V$ are the unique projection morphisms of the binary product.

In a general category

A loop graph object in a category $\mathcal{C}$ is a loop digraph object$(V, E, s:E \to V, t:E \to V)$ with a morphism $i:E \to E$ such that