= Solution
Choose an integral <Minimal Weierstrass equation> whose discriminant is a $p$-adic unit. Given $P=[X:Y:Z]\in E(\mathbb Q_p)$, scale the coordinates so that $X,Y,Z\in\mathbb Z_p$ and at least one is a unit. Define
$$
\varphi(P)=[\bar X:\bar Y:\bar Z]\in\widetilde E(\mathbb F_p).
$$
Another primitive representative differs by a unit scalar, so the projective reduction is well defined. The reduced cubic is smooth by <good reduction>.
To prove the <group homomorphism> property, view this smooth projective cubic with its section $O$ as the integral elliptic model $\mathcal E/\mathbb Z_p$. Its addition law is defined over $\mathbb Z_p$, by adding degree-zero <divisor classes>, and thus is a regular morphism $\mathcal E\times\mathcal E\to\mathcal E$. Properness identifies $E(\mathbb Q_p)$ with the integral sections just described. Reducing the addition morphism is the <group operation> of the smooth special fiber. Consequently
$$
\boxed{\varphi(P+Q)=\varphi(P)+\varphi(Q),\qquad\varphi(O)=\widetilde O.}
$$
This is the <good-reduction map respects elliptic addition> argument; it covers tangent, vertical and infinity cases where reducing a single affine slope formula would involve division by a nonunit. Smoothness also makes reduction surjective: at each reduced point a partial derivative is a unit, and the <Hensel lemma> lifts that point in a local affine chart.
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