 Monotonicity of relation composition

If $R\subseteq R^{\prime }$ and $S\subseteq S^{\prime }$, then $R\circ S\subseteq R^{\prime }\circ S^{\prime }$. A witness joining a pair through $S$ and then $R$ also joins it through the larger relations. If both $R$ and $S$ are reflexive relations on the same set, inserting an identity step gives $R\cup S\subseteq R\circ S$, so monotonicity also gives $(R\cup S{)}^2\subseteq (R\circ S{)}^2$.