Let $f:X\rightarrow Y$ be a separable isogeny of abelian varieties and $K$ be its kernel. Let $\widehat{f}:\widehat{Y}\rightarrow \widehat{X}$ be the dual map and $K'$ the kernel of this. It is known that $K'$ can be identified with the Cartier dual $\operatorname{Hom}(K,\mathbb{G}_m)$ e.g. Mumford's Abelian Varieties Section 15 Theorem 1.
I found Mumford's proof lacking in intuition for me. Why might one believe this result to be true and can someone provide geometric intution for this result?
Going from definition does not bring much enlightenment and neither does viewing this from the context of elliptic curves as far as I can tell. One way to view this is to given in Polishchuk's Book where it is shown that $\operatorname{Ext}^1(X,\mathbb{G})=\widehat{X}$. In that case, one can deduce the claim from a LES of $\operatorname{Ext}$-groups. Of course, my suspicion is that the idea is the same that in Mumford's (just phrased in a modern way) and so I have not studied Polishchuk's proof as carefully.