"Our institute's professors are erudite."
Translation:Han yejHaDmaj 'a'ghenpu'.
-maj and -Daj are both possessive suffixes. You cannot use two possessive suffixes on the same noun. I'm not sure what "our his institute" would mean anyway.
I wonder if you might have thought you needed -Daj to get the idea of the English -'s, but that would not be a correct understanding. -Daj should be translated into the sentence as "his", "her", or "its". The -'s is represented in Klingon by simply placing the nouns adjacent to each other. In other words, when you stick two nouns next to each other then the first describes the type of the second. So tlhIngan Hol is "a Klingon type of language". baS yan is "a metal kind of sword". lIy ghop is "a Lee kind of hand" (or "Lee's hand"). And yejHadmaj 'a'ghenpu' are "our institute kind of professors" (or "our institute's professors").
Klingon does not represent English 's with apposition.
Apposition is when you've got two noun phrases side by side, and one phrase is being identified as the same as the other phrase. SID Haq Qel, mara The doctor, Mara, performs surgery on the patient. When you put Qel and mara next to each other, you're putting them in apposition: it tells you that the doctor and Mara are one and the same person.
In Klingon, English 's is usually translated with what is known as a noun-noun construction. It is positionally the same as apposition: noun phrases are put side by side. But the Klingon noun-noun construction is a genitive construct: it says that the first noun phrase modifies the meaning of the second noun phrase. jIHvaD nuH pegh yIja' Tell me the secret of the weapon; tell me the weapon's secret. You know this is not apposition simply by context: it makes no sense as Tell me the secret, the weapon.
Thanks for pointing out my mistake. I was indeed using the lay definition of apposition, but since this is a grammatical discussion I should have paid attention that it might be interpreted to mean "having the same referent". I'll edit the word "apposition" in my previous reply to refer to adjacent nouns instead.