restriction of scalars and module homomorphisms

Let \(R,S\) be rings, \(\varphi: R \rightarrow S\) a ring-homomorphism and

\begin{align*} f: M \rightarrow N \end{align*}

a module-homomorphism for \(S\)-modules \(f\)

Then \(f\) is a module homomorphism for the restriction of scalars \(R\)-modules \(M,N\)

it remains to show linearity: suppose \(r \in R\), then

\begin{align*} f(r \cdot_R m) =& f(\varphi(r) \cdot_S m) \\ =& \varphi(r) \cdot_S f(m) \\ =& r \cdot_R f(m) \end{align*}