Proposition 9

Preserving the above notation, let $f$ be a homomorphism of $M$ into a magma such that $f(u_\alpha) = f(v_\alpha)$ for all $\alpha \in I$. Then $f$ is compatible with $R$.

Let $T$ be the equivalence relation associated with $f$. Then $u_\alpha \equiv v_\alpha \pmod{T}$ for all $\alpha \in I$ and $T$ is compatible with the law on $M$, hence $T$ is coarser than $R$; this proves the proposition.