I put "the fighter" instead of "the warrior" and it was wrong, but "suv suvwi'pu'" means "fighters fight"?
Yes and no.
Yes, it has a prefix to show the subject and the object -- but the prefix for "subject = third person singular (he/she/it); object = third person singular (him/her/it)" is the null prefix or empty prefix: nothing at all.
So Suv has an empty prefix, which looks the same as no prefix (as if you had forgotten to add a prefix).
Suv can thus mean not only "he fights" but also "he fights him" or even "he fights them" and "they fight" and "they fight them". (But not "they fight him" -- that combination of subject and object does have a prefix.)
I accidentally left off the "-s" at the end of "fights" in "The warrior fights his foe." It reported the typo and corrected it to "The warriors fight his foes", which is clearly an incorrect translation of this sentence. What gives?