Learn Polish: Musi pan pokazać oryginał, abyśmy sprawdzili dokument. - You must show the original so that we check the document.

When a purpose clause has a different subject than the main clause (the man shows, but we check), Polish requires aby, by, or żeby combined with the past tense form of the verb. We take the 3rd person plural past form (sprawdzili) and attach the personal ending for "we" (-śmy) directly to the conjunction to make abyśmy.

Can I just say aby sprawdziliśmy instead?

No, that is incorrect. In conditional and purpose clauses, the personal ending (like -śmy) must attach to the conjunction, creating abyśmy. The verb itself must be left in the 3rd person past tense form (sprawdzili).

Is there a difference between using abyśmy and żebyśmy in this sentence?

Grammatically they function exactly the same way, but abyśmy belongs to a more formal, written, or professional register. Because this sentence is a formal instruction using pan, abyśmy matches the official tone much better than the everyday żebyśmy.

Both oryginał and dokument are direct objects. Why haven't their endings changed?

They are indeed in the accusative case because they are direct objects. However, because both are masculine inanimate nouns, their accusative forms look exactly identical to their basic nominative dictionary forms.