Learn English – What part of speech is “down” in “down went the Titanic”

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.

In the sentence "Hiking is fun," the word "fun" is an adjective. It's the same type of construction as, for example, "My mother is tall."

The word "fun" can be a noun as well as an adjective, and the verb "is" can link two nouns ("My mother is a teacher"), so I can see where you might be confused. But in the sentence "Hiking is fun," most English speakers would understand "fun" as an adjective, almost as if it were shorthand for "Hiking is a fun thing to do."