There's a fair bit of disagreement on that. It's been called a pronoun 
(the traditional label), a determiner, a predeterminer, and an adverbial 
quantifier that I'm aware of, and I'm sure there are some analyses I'm 
unaware of. How you analyze it will depend a lot on your theoretical 
assumptions.

Like Sergio, I see its form as a determiner (or determinative if you use 
the CGEL terminology). I'm also inclined to see it as _not_ a 
constituent of the subject, even though semantically it quantifies it. 
Notice that if we add an auxiliary, "all" naturally sits in the 
post-auxiliary position:

The happy teachers and their smiling students have all enjoyed the 
grammar party.

For that reason, I'd say it functions as a quantifier, as an adjunct 
within the VP. That said, there are a number of different analyses, in 
both the generative and the GPSH/HPSG traditions which do take it as a 
constituent of the subject, at least when it's not sitting in 
post-auxiliary position, but it remains an adjunct (albeit of the 
subject) pretty much any way you slice the analysis.

Regards,

Karl

On 1/11/2018 1:07 PM, Sergio Pizziconi wrote:
> Hello Scott,
> I would say it is a determiner.
> If it were in a more canonical position, but clearly less probable 
> (statistically speaking), such as "All the happy teachers and all 
> their smiling students....", its nature of determiner would be apparent.
> Sergio Pizziconi
>
> 2018-01-11 21:51 GMT+01:00 Scott Woods 
> <[log in to unmask] 
> <mailto:[log in to unmask]>>:
>
>     Hi All,
>
>     In the following sentence <The happy teachers and their smiling
>     students all enjoyed the grammar party> how would you characterize
>     the word <all>?
>
>     What is its function and what would you call it?
>
>     Thanks,
>
>     Scott Woods
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