composition of group homomorphisms as group homomorphism

Let \(G_1,G_2,G_3\) be groups and \(\varphi_1: G_1 \rightarrow G_2\) and \(\varphi_2: G_2 \rightarrow G_3\) be group-homomorphism. Then the composition

\begin{align*} \varphi_2 \circ \varphi_1: G_1 \rightarrow G_3 \end{align*}

is also a group homomorphism

corollary of composition of monoid homomorphisms as monoid homomorphism