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)