The check should be made payable to the customer. The customer sitting across from you can simply endorse it on the back.
The customer mailing a check drawn on a different bank for deposit in your bank should also make the check payable to himself. He can then affix a restrictive endorsement beginning "For Deposit Only." Thereafter, they can affix the name of the institution and even the specific account number they want before they sign their name.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.