Group homomorphism/Categorial properties/Fact

Let ${}F,G,H$ denote groups. Then the following properties hold.

1. The identity

   $\operatorname {Id} \colon G\longrightarrow G$

   is a group homomorphism.

2. If ${}\varphi :F\rightarrow G$ and ${}\psi :G\rightarrow H$ are group homomorphisms, then the composition ${}\psi \circ \varphi \colon F\rightarrow H$ is a group homomorphism.
3. For a subgroup ${}F\subseteq G$, the inclusion ${}F\hookrightarrow G$ is a group homomorphism.
4. Let ${}\{e\}$ be the trivial group. Then the mapping ${}\{e\}\rightarrow G$ that sends ${}e$ to ${}e_{G}$ is a group homomorphism. Moreover, the (constant) mapping ${}G\rightarrow \{e\}$ is a group homomorphism.

Write a proof