Here is a sentence: 有時我唔開心會喊。之後我就心情好啲。
I don't understand the function of the word 就. In my mind, it is supposed to be 嘅, for possessive, considering the context and that 心情 (feelings) would be a noun. Why is 就 used here?
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You think there is something wrong with this sentence because 我嘅心情 (my mood) is a noun; 好啲 (better) is an adjective. A verb is seemingly missing in this sentence
It would be easier to rewrite the sentence as '有時我唔開心會喊，之後我嘅心情就會好啲。' (Sometimes I cry when I am unhappy. Afterward, my mood would become better) or '有時我唔開心會喊，之後我就會開心番' (Sometimes I cry when I am unhappy. Afterward, I would become happy again)
However, if you must use 之後我就, you have to treat 心情好啲 as a verb phrase
'有時我唔開心會喊，之後我就(會變得)心情好啲。' --> Sometimes I cry when I am unhappy. Afterward, I (would become) in a better mood
This seems somehow incomplete, unfinished (although, I know nothing of Guangdonghua):
(When I'm unhappy), (then) I scream,
(after screaming) my mood is (then) better.
This is 2 instances of the subjunctive, each with 2 clauses:
when ... then, or condition ... result
shout, shout, let it all out!
This Cantonese sentence is perfectly fine (judging from a native Cantonese speaker).
You can understand the word 就 as 'turn into the state of ...'
and '心情好啲' being an adjective phrase.
So the sentence means, 'after that, I turn into the state of "being in a better mood"',
which could be rephrased into 'after that, my mood becomes better.'
In Cantonese and in Mandarin, 就 has more or less the same syntactic behaviour and function. For my convenience, I use Mandarin sentences here as examples (it is easier for me to type in Hanyu Pinyin).
In the construction "X 就 Y", 就 indicates that X is a minimum value that validates Y. For example:
For 就 to work properly, X must be interpretable as a minimum value in the first place. Let's compare (3a) with (3b):
In normal circumstances, "几天不吃饭" is difficult to be interpreted as a minimum value, though it is not impossible. Indeed, it is more common to see people use 才（or 至 in Cantonese) instead of 就 in this case:
Incidentally, 才 indicates that X is a maximum value that validates Y.
The opposition of minimum/maximum values can take the form of quantity (less/more), time (early/late), scale (low/high), sequence (first/last)... etc. In the example below, it takes the form of sequence:
A woman is talking to two kids. They are twin brothers:
The woman: 你们谁是哥哥？
Brother A: 我就是。(The first one to answer)
Brother B: 你乱讲！我才是哥哥。(The last one to answer)