Syntactic Analysis – What Part of Speech is ‘Fun’ in ‘Hiking is Fun’?

Both you and your friend are incorrect; sorry.
But that's not your fault; you're at a disadvantage,
if you take your definitions of "part of speech" from English books.
They're hopeless; pay no attention to them.

Here is a proximal deictic locative predicate in the sentence

• I am here.

It does not modify the verb am.
It does not modify anything, in fact.
(Be) here is the Predicate in the sentence.

The logical form is

• HERE (I)

The am is indeed an auxiliary verb, meaning -- if anything -- 'be located (at)'.

Executive Summary: Calling something an "adverb" is a confession of ignorance.

Indifferent is not the correct word to describe the way native English speakers regard parts of speech; it's more like flexible:

In the unambiguous expression:

the children enjoyed the fun and games

Fun is a noun.

In the unambiguous expression:

the children enjoyed the fun games

Fun is an adjective.

In the ambiguous expression:

it's fun

The larger context will determine the part of speech:

Fun is renaming it as a predicate noun.

or

Fun is describing it as a predicate adjective.

As long as people can perceive, use and interpret parts of speech properly, it is not so important to label them. We teach our children to label parts of speech so that they can learn to speak, read and write properly. As instructors, the labels remain supremely relevant to us, but once our students can speak, read and write, the labels quickly become less relevant to them. Those who never discuss parts of speech, tend to forget the significance of the labels, because they don't need the labels to communicate.