nouns - What word class/part of speech is " the doctor's " in " the doctor's office "?

I need to be able to identify the word classes/parts of speech of each word in a sentence such as:

"He walked into the doctor's office."
Is it a possessive determiner? A possessive adjective? A possessive noun?

This wikipedia page indicates (without reference) that it can be called a determiner: https://en.wikipedia.org/wiki/Possessive_determiner. I see the parallel with "The doctor called him; he walked into her office," where "her" - which is a possessive determiner - serves an almost identical function as "the doctor's". It's determining the specificity of the office, much like the determiners "the office" or (that office). However, most pages I've found on determiners don't mention this type of formation, and they present determiners as a closed word class (which logically can't include all cases of "[noun]'s").

"Doctor" in the sense referred to here is a noun, and the 's commonly shows possession, but in the case of "the doctor's office" the word "doctor's" isn't a noun in its own right, is it? Without the head noun "office", it doesn't signify any specific object, merely the quality of belonging (as in "the doctor's house", "the doctor's garden" etc).

As it's modifying/adding information about the office, is it an adjective in this case?

Does it depend on whether "the" is considered to be referring to the (specific) doctor or to the (specific) office?

Thanks for any help!