Answer:
The UCC places the responsibility for obtaining the endorsement on the bank of first deposit (that's you). The only opinion that matters on endorsements is yours; i.e. if the paying bank returns the item "not endorsed as drawn" or there is a claim that the payee never received the funds you will have to deal with it.
In short, if you want to require them to properly endorse the checks in their name you may certainly do so and all you have to do to support that requirement is say, "Because we say so."
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