Let's set the scenario that we see most often:
Account styled -- Cathy Customer
Check payable to -- Cathy's Catering
Now, if you cash the check, or deposit the check to Cathy's account; how do you know that there is not another Cathy's Catering in New York? Maybe someone just looked up the address in the phone book and sent a check. Of course, the lenders generally say "We know who we are dealing with" ----- until there is a problem!
You can protect yourself and keep your options open by styling the account Cathy Customer dba Cathy's Catering, get the resolution, and have the customer deposit all checks to Cathy's Catering and then cash her own check. Also, the customer can still order their checks with only their name (Cathy Customer) if they so desire. This is what we do and the customers generally accept it once they realize that they do not have to put the business name on the checks --- most of them order two checkbooks, one with and one without.
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NOLA is my Beach!