My book (Silverman's Arithmetic of Elliptic Curves) says on page 134, Chapter 4, section 7, in the discussion related to formal groups in characteristic $p$ that -
Theorem $7.4$: Let $K$ be a field of characteristic $p > 0$, let $E_1/ K$ and $ E_2/K$ be elliptic curves, and let $\phi : E_1 \mapsto E_2$ be a nonzero isogeny defined over $K$. Further, let $f: \hat{E_1} \mapsto \hat{E_2} $ be the homomorphism of formal groups induced by $\phi$. Then $$ deg_{I} (\phi) = p^{ht(f)}$$
My question- How does an isogeny $\phi$ on elliptic curves induces a homomorphism of corresponding formal groups?
I think maybe it has something to do with how a formal group can be associated with an elliptic curve over a field of fractions $K$ of a complete local ring $R$ (on page 123, section 3) but here in the context of the theorem
(1) our ring need not be local and complete. And (2) even if we assume somehow that formal group for $E$ can be defined I do not see how $\phi$ will induce $f$?
I'd appreciate it if someone could explain this to me. Thank you!