composition compatible Relation

Let \(\mathcal{C}\) be a locally small category and \(R\) a Relation on each hom-set. Then \(R\) is said to be composition compatible, if for \(f_1,f_2 \in \mathrm{Hom}_{\mathcal{C}}(A,B)\) and \(g_1,g_2 \in \mathrm{Hom}_{\mathcal{C}}(B,C)\) with \(f_1 \sim f_2, g_1 \sim g_2\) it follows, that

\begin{align*} g_1 \circ f_1 \sim g_2 \circ f_2 \end{align*}