Define an entanglement breaking channel $\Phi$ as a channel (CPTP map) of the form $$\Phi(\rho) = \sum_a \operatorname{Tr}(\mu(a)\rho) \sigma_a\tag A$$ for some POVM $\{\mu(a)\}_a$ and states $\sigma_a$.
It is mentioned e.g. in (Horodecki, Shor, Ruskai 2003) that $\Phi$ is entanglement breaking iff it "breaks entanglement", that is, is such that $$(\Phi\otimes I)\Gamma \quad\text{ is separable for every state } \Gamma.\tag B$$
This equivalence is proved, I think, in pages 5 and 6 of the above reference, but I can't quite follow the exposition there. In particular, the proof that (A) implies (B) seems to rely on expressing the action of $\Phi\otimes I$ on $\Gamma$ via a partial trace involving some operators $E_k$ which however are not defined (there might be a typo somewhere in the text, I'm not sure).
What are good ways to prove the equivalence between (A) and (B)?