= Solution
A <flat module> is one for which tensoring preserves injections. This is equivalent to exactness of the tensor functor, since the <tensor product of modules> is always right exact: its universal balanced maps identify tensoring a cokernel with the cokernel of the tensored map.
Suppose $M$ is <flat>. For every inclusion $N'\hookrightarrow N$, the map $N'\otimes_AM\to N\otimes_AM$ is injective. The detection property proved above therefore makes its character-dual map surjective. Under the character <Tensor-hom adjunction>, this is exactly
$$
\operatorname{Hom}_A(N,M^*)\twoheadrightarrow\operatorname{Hom}_A(N',M^*).
$$
Every map from a submodule into $M^*$ thus extends, proving
$$
\boxed{(1)\Longrightarrow(2).}
$$
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