= Solution
We use the basic <dual isogeny> identities in the <endomorphism ring of an elliptic curve>, extending the dual of the zero map to zero:
$$
\widehat{\phi+\psi}=\widehat\phi+\widehat\psi,\qquad
\widehat{\phi\psi}=\widehat\psi\,\widehat\phi,\qquad
\widehat{[n]}=[n],\qquad
\phi\widehat\phi=\widehat\phi\phi=[\deg\phi].
$$
Here $[n]$ denotes the multiplication-by-$n$ <endomorphism>, and the degree of the zero map is set to zero. To see the source of additivity, identify $E$ with its <Picard group> of degree-zero line bundles using the origin. Dualization is pullback on this group. Pullback of a degree-zero line bundle along the sum of two homomorphisms is the tensor product of its pullbacks along those homomorphisms, giving additivity; functoriality of pullback reverses composition. The last identity is the defining degree identity for a dual isogeny. Every nonzero elliptic-curve endomorphism is an isogeny.
The natural inclusion $\mathbb Z\hookrightarrow\operatorname{End}(E)$ remains injective even in positive characteristic: for nonzero $n$, the map $[n]$ has degree $n^2$ and is not zero. Therefore an equality of multiplication endomorphisms uniquely specifies its integer coefficient.
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