In a proof in Milne's note "Abelian Variety" (top on p.52), I saw an equality: $ \mathrm{Ker}(\beta)(l) = \mathrm{Coker}(T_{l}(\beta))$, here $\beta$ is an (separable) isogeny of an abelian variety $A/k$, $l$ is a prime number different from the characteristic of the base field $k$, $(l)$ means the torsion points of order a power of $l$, $T_{l}(\beta)$ is the induced map of $\beta$ on the Tate module of $A$.

I can't figure out why this equality holds. Do we have a natural map for these two groups?