homomorphism ring ~

Suppose $R,R'\text{ dan }R''$ are rings and mappings $\varphi : R \rightarrow R'\text{ and }\varphi ' : R'\rightarrow R''$ both are homomorphisms. Then show that $\varphi '\varphi : R\rightarrow R''$ is also a homomorphism!
Just show directly that the requirements are satisfied by the composite function $\varphi ' \varphi$. For instance we have, for all $x,y \in R$, $\varphi ' \varphi(x+y)=\varphi'(\varphi(x) + \varphi(y)) = \varphi ' \varphi(x) + \varphi ' \varphi(y)$.