What is an inward check? What is a debit on-us check?
"Inward check" is probably what most bankers refer to as an "in-clearing check," which is a check that arrives at the bank it's drawn against along with other checks that were deposited at other banks. The process of in-clearing is the way checks "arrive home," and the bank has to decide whether those checks are "good" for posting to the accounts on which they are drawn. Contrast "in-clearing" with "over-the-counter" checks, which are deposited at the very same bank they are drawn against.
The term "debit on-us" is often used by a bank to indicate that the check in question was presented for payment across the bank's teller counter. The term "on-us" refers to checks that are deposited to the same bank they are drawn on.
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