Dear Bera
Wow thank you for your response.
I find the tarot card system is somewhat different to connection to humours, it appears as if the Swords suit is more similar to the Choleric in temperament, yet it is linked to Air . It appears too decisive, rigid and just too solidified to be Air in my view. I can definitely see how it may be linked to Je, as Auburn mentions the sword in both the Te and Fe myths as being a symbol of divine agency. However the Swords suit is too intellectual, logical and almost harsh in my view to be linked to Fe. This is why I feel Te or Fe could be blended with another element perhaps.
The Wands suit appears to be linked to Fire, but it has a combination of Choleric and Sanguine traits. The myth of Ne which is the Puer Eternus has the ability to distort or alter reality, which sounds a lot like a magic wand. Intuition seems traditionally linked to the fire element as well. There is also a spirit of adventure there and a desire to explore, which we can find in both Se and Ne types.
The Cups suit definitely fits the Water element in astrology as being tied to Feeling, but the Phlegmatic temperament isn't nearly as emotional, I'd rather say the Melancholic is emotional.   Though I would link it to Ji, due to its emotionality I find it difficult to really assign it to Ti, which is absent in ethical and emotional awareness.
The Pentacles seems to align with Sensation, and by association with earth being a yin or passive element, it is most likely linked to Introverted Sensation, also tying in with the myth of the earth as I stated before. This may apply to Pi as well, however I find this problematic when it comes to Ni, which is by nature non-concrete and abstract, which is in direct contrast to the earth element.
I am surprised that Jung would link Water to Thinking, I always thought Thinking was linked to Air. Perhaps it makes sense when you tie it to the Phlegmatic which is by nature rather dispassionate. But the traditional association with water in the tarot system as well as in astrology seems to conflict with this a lot, as it is tied to feelings and decisions revolving emotions, ethics and moral values.
Another thing that confuses me is that the Sanguine Temperament is linked to the Air element, but when I read about the air zodiac signs it only partially correlates, the correlation between air and thinking is not the same as the correlation between the sanguine and the air element. Sanguine's are believed to be more emotionally expressive, so it confuses me that it would be linked to Air, which is according to astrology and the tarot card system quite dispassionate and emotionally detached.
Also while I do like Auburn's associations, I find the Ji Melancholy connection to the earth element a bit problematic, because he mentions Ji types as not necessarily being very productive or practical, but rather idealistic. This does not seem to align with the earth element in either the tarot deck or astrology. its probably only linked to the Melancholy temperament subtracted from its element, because the traditional description of Melancholy that I've seen on a catholic website about the four temperaments aligns very much with the Ji traits.
I will share what this website says about the four temperaments. https://www.fisheaters.com/fourtemperaments.html
Sanguine
Self-composed  Not given to worry  Liberal
Tends to follow rather than lead  Cordial  Peaceable
Talkative  Not averse to change  Adjusts easily
Tends to prefer informality  Aware of surroundings  Impetuous
Impulsive  Lacking in perseverance  Lacking in initiative
Prone to carelessness, hedonism, flightiness, and lust
Humor: Blood   Element: Air   Season: Spring
Qualities: Hot and Wet  Planet: Jupiter
Sanguine appears to link to the Pe metabolism very well, especially in terms of its hedonism, flightiness and lust (the lust could be linked more to Se lust, while the flightiness could link more to Ne)
Choleric
Self-composed  Not given to worry  Persuasive  Independent
Rarely shows embarrassment  Tends to lead rather than follow
Persistent  Insistent  Decisive  Dynamic  Impetuous  Impulsive  Touchy
Prone to hypocrisy, deceit, pride, and anger
Humor: Yellow Bile  Element: Fire  Season: Summer
Qualities: Hot and Dry  Planet: Mars
Choleric does appear to link to Je very well, when you look at pride it reminds of how Fe seeks to control society in terms of managing its pride and shame.
Melancholic
Sensitive  Intuitive  Self-conscious  Easily embarrassed
Easily hurt  Introspective  Sentimental  Moody
Likes to be alone  Empathetic  Often artistic
Often fussy and perfectionist  Deep
Prone to depression, avarice, and gluttony
Humor: Black Bile  Element: Eartth  Season: Autumn
Qualities: Cold and Dry  Planet: Saturn
The Melancholic temperament aligns very well with the Ji metabolism, as both are prone to being fussy and perfectionist, in some cases prone to depression as well. The empathetic nature and sentimentality would probably align more with Fi, while the other traits of avarice, introspection and desire for solitude could apply to both Fi and Ti.
Phlegmatic
Peaceful   Easy-going  Deliberative   Faithful   Reliable
Relatively unaffected by environment  Reserved  Distant
Slow in movement  Constant in mood  Not prone to worry
Prone to stagnation and sloth
Humor: Phlegm  Element: Water  Season: Winter
Qualities: Cold and Wet  Planet: Moon
The Phlegmatic temperament appears to fit the Pi metabolism very well, as it is prone to a certain inertia and has a rather viscous flow, similar to the bodily humour of Phlegm which is rather viscous. It is generally dependable and reliable, but is far less fussy and perfectionistic about the details of things.
So if we only look at the four temperaments before adding in the four elements, I'd say Auburn has done an excellent job of correlating them. I think I was just looking for more specifics for certain functions like Ti and Te, because the mythology of those two functions don't really seem tied to a specific element, but they are definitely tied in very well with their metabolism and temperament.
 

Harry Potter House theory
So I think Harry Potter Houses are an interesting system that I followed. The four elements don't really correspond to the Arcane types with Harry Potter, its more about the general temperaments.
Hufflepuff would be associated with Introverted Perception and are Phlegmatic, they very much fit the traits of the Senex or Crone. Hufflepuffs value loyalty, hard work and kindness, so they would be probably be a Pi-Fe type.
Ravenclaw would be associated with Introverted Judgement and are Melancholic, fitting the traits of the Prince/Princess. They value intelligence, curiosity, have high standards and are quite idealistic, I would say they are a Ti-Pe type.
Gryffindor would be associated with Extroverted Perception and are Sanguine, fitting the traits of the Explorer and Adventurer. They value bravery, daring, adventure and seem to value fighting for the underdog and standing up for individual beliefs. Thus I would say they are Pe-Fi.
Slytherin is associated with Extroverted Judgment and are Choleric, fitting the traits of the Ruler and Articulator. They value cunning, are intelligent, resourceful, good at strategy and are typically aspiring leaders. I would say they are Te-Pi.
Also interestingly the four founders genders could have played a role in their temperaments, because when looking at the four elements and tarot suits, the male elements are usually hot elements (air and fire) and this also fits the Sanguine and Choleric. So Godric Gryffindor would be Sanguine, while Salazar Slytherin would be Choleric.
The female founders and their respective elements are cold ones (earth and water) and this also fits the Phlegmatic and Melancholic. So Rowena Ravenclaw would be Melancholic, while Helga Hufflepuff would be Phlegmatic.