= Solution
An <atomic model> is a model in which the type over the empty set of every finite tuple is isolated. Let $\bar a\mapsto\bar b$ be partial elementary and take $c\in M$. Choose a formula $\theta(\bar x,y)$ isolating $\operatorname{tp}(\bar a,c)$. Since $M\models\exists y\,\theta(\bar a,y)$, elementarity gives $N\models\exists y\,\theta(\bar b,y)$. Any witness $d$ realizes the isolated joint type, so $\bar a,c\mapsto\bar b,d$ is partial elementary. This is the <one-point extension between atomic models>.
Alternately applying this extension property in the two directions to enumerations of countable atomic models constructs a <back-and-forth method>[back-and-forth] isomorphism.
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