restriction of a module homomorphism to the image as module homomorphism

Let \(R\) be a ring, \(\varphi: M \rightarrow N\) a module-homomorphism and \(L \subseteq N\) a submodule such that

\begin{align*} \mathrm{im}(\varphi) \subseteq L \end{align*}

Then the map

\begin{align*} \varphi': M \rightarrow L \\ m \mapsto& \varphi(m) \end{align*}

is a welldefined module homomorphism

Properties of the module homomorphism are inherited by \(\varphi'\), hence it follows from the definitions