# A question regarding tensor product and isogenies of elliptic curves

Let $E_1$ and $E_2$ be elliptic curves and $T_l(E_i)\cong \mathbb{Z}_l \oplus \mathbb{Z}_l$ the $l$-adic Tate module.

Given $\varphi \in Hom(E_1,E_2)$ this induces $\varphi_l \in Hom(T_l(E_1),T_l(E_2))$ and the map $\varphi \mapsto \varphi_l$ is injective.

Why is the natural map $Hom(E_1,E_2)\otimes \mathbb{Z}_l \longrightarrow Hom(T_l(E_1),T_l(E_2))$ being injective a stronger statement?

(This appears in Silverman's Arithmetic of elliptic curves p89)

$\hom(E_1,E_2) \longrightarrow \hom(E_1,E_2) \otimes_{\mathbb Z} \mathbb Z_l \longrightarrow \hom(T_l(E_1),T_l(E_2))$
Since $\hom(E_1,E_2) \otimes \mathbb Z_l$ is strictly bigger than the image of $\hom(E_1,E_2)$, it is better to know that each map is injective than to know that their composition is injective.