Let $K$ be a field, and $\phi: E_1\to E_2$ be an isogeny of elliptic curves over $K$. Given a prime $\ell$ different from the characteristic of $K$, $\phi$ induces an injection $T_\ell \phi: T_\ell E_1\to T_\ell E_2$. It is easy to check group theoretically that the cokernel of $T_\ell \phi$ is isomorphic to the $\ell$-primary part of $\ker\phi$. Can this isomorphism be made canonical?
Note: Over $\mathbb{C}$, this can be written down pretty straightforwardly. Let $E_i=\mathbb{C}/\Lambda_i$, where $\Lambda_i$ is a lattice. We make identify $\Lambda_1$ as a sublattice of $\Lambda_2$ via the isogeny. This gives an short exact sequence
$0\to \Lambda_1 \to \Lambda_2 \to \Lambda_2/\Lambda_1\to 0,$ where $\Lambda_2/\Lambda_1$ is naturally identified with $\ker{\phi}$. Tensoring with $\mathbb{Z}_\ell$ gives us $ 0\to T_\ell E_1 \to T_\ell E_2 \to (\ker \phi)[\ell^\infty]\to 0.$
But I am having trouble to do it over an arbitrary field.