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I have tried to make the title catchier. - @JukkaKohonen : $\varphi(e_1)= \text{im}(\varphi|_{e_1})$, so $\varphi(e_1)$ is the image of $e_1\subseteq V_1$ under $\varphi$. This is a subset of $V_2$. Now $E_2$ is a collection of subsets of $V_2$. Whence the requirement $\varphi(e_1)\in E_2$. Does that make sense?