Hi there, so this is my own correlation I made between the Enneagram types and their correlation to Cognitive Functions based largely on the descriptions of both. A disclaimer: I know not everyone's enneagram type coincides perfectly with their Cognitive Type, but I just did this comparison for my own interest.
Enneagram Type 1 The Reformer: Te, Fe, Unseelie, Directive
This type seems to correlate largely with both Je functions. I can see in both Fe and Te a need for control and order in the external environment. While Te is more emotionally detached and cerebral and keeps their passions in check, Fe shows the moralistic, nagging and need for improvement from an idealistic level. I also saw a correlation between Type 1 and Directive and Unseelie attitudes. It seems like a largely Disagreeable or Emotionally Guarded type.
Enneagram Type 2 The Helper: Fe, Adaptive
This type seems to correlate mostly with Fe, as they are attuned to interpersonal dynamics and want to help others and the community or even the world at large. They are emotionally expressive outwardly, and I think this ties in with Adaptive Fe especially, since Adaptiveness is linked to doing kind deeds, being a martyr or self-sacrificial and seeing themselves as a nurturer or provider, but there are also traces of it in the Adaptive Ti-Fe where the outward persona seems more like that of a Two.
Enneagram Type 3 The Achiever: Fe, Te, Unseelie
This type also seems spread across more than one cognitive function. I initially found it to be somewhat linked to Te, which is meritocratic and wants to climb the corporate ladder. However I also see this in Fe who wants to achieve their goals and are quite ambitious. I am unsure of the emotional attitude of Type 3, but I did see some correlation with the vanity, self-obsession and desire for status in the Unseelie attitude.
Enneagram Type 4 The Individualist, Fi Seelie and Unseelie
This type largely corresponds with Fi. Both Fi and Type 4 are emotionally sensitive, individualistic, private, melancholy and seek to express their individuality in unusual ways. A connection to the past and nostalgia seemed to feature subtly in Si as well, and also perhaps a holding of grudges and refusing to let go of the past. I can see Type 4 expressed in both the Seelie and Unseelie attitudes, where the Seelie Type 4 is more emotionally vulnerable and deeply empathetic and feels the injury of living beings, while the Unseelie Type 4 is haughty, elitist and has refined tastes, while also showing a jealousy or envy of others. The darker side of the Unseelie manifests in Type 4 as well where they are interested in the morbid and macabre.
Enneagram Type 5 The Investigator: Ti
This type largely corresponds with Ti. There is a strong emotional detachment, reliance on logic and a fear of intrusion of privacy. This type is the most analytical and is cool, calm and collected. It also has traits such as being bizarre and idiosyncratic, which may cross over with Fi, but this eccentricity would probably manifest in a more analytical, emotionally detached manner. There was also a bit of a correlation with Type 5's desire for knowledge and an avarice hoarding with Si. Type 5 seems to correspond more with the Directive attitude with Ti dominant types, but with the Adaptive type we see its dispassionate interiority and scrutinizing side as being more like a Type 5.
Enneagram Type 6 The Loyalist: Si, Ni
This type corresponds with both Pi functions when it is under stress: Si and Ni. I almost felt like it was exactly the same as Si under stress, especially when it comes to Si not trusting their own judgments and needing an external belief system or explanation to cling onto, of which they become the most loyal and devoted. I found Ni to be the more intuitive, abstract parallel of Si and I found the catastrophic thinking and apocalyptic visions of Ni to resonate strongly with Type 6's fearfulness and dark pessimism. I couldn't really find an emotional attitude to correlate Type 6 with, but it does seem to also correlate slightly with Fe in regards to being interdependent and connected to the collective.
Enneagram Type 7 The Enthusiast: Se, Ne
This type corresponds with both Pe functions: Ne and Se. Type 7 is the explorer and seeker of novel experiences in both the intuitive abstract world, as well as the sensual, concrete world. I found it to be more strongly connected with Ne which is generally more optimistic and looks on the bright side compared to Se which is generally more realistic and may feel stuck in an uncomfortable situation for longer periods of time. But Se also adapts and adjusts to each moment and can improvise so well, which is a strong Type 7 characteristic, and one which I personally identify with as an SeFi type. I couldn't find any correlation with an emotional attitude for Type 7, but I do know they try to avoid pain and an unhealthy Seven can even become selfish, insensitive and jaded, so this may point to the Unseelie side a bit. But since it stretches across two cognitive functions, it could easily be any of the four emotional attitudes.
Enneagram Type 8 The Challenger or Leader: Te, Fe, Directive, Unseelie
This type initially struck me as only being Te, being self-sufficient, tough-minded, insensitive to their own weakness and diligent, but upon further inspection I found it correlates strongly with Fe as well, particularly because Type 8's are seen as heroes and protectors who take on insurmountable challenges, which I found to correlate very well with the Myth of the Fe Protagonist. In terms of Emotional attitudes, I found Type 8 to correlate with the Directive Fe quite strongly. For the Te-Fi spectrum, it probably correlates more with Unseelieness.
Enneagram Type 9 The Peacemaker: Si, Ni
When I looked at the actual description of the Pi Worldview type, it made perfect sense that it would fit Type 9 especially because it is stable, calm, unenthusiastic and has a generally Phlegmatic temperament. In Ni we see how there is a ceasing of separation between matter and spirit and Ni types generally seek an awakening or enlightenment which correlates with the description of Type 9. In Si there is a sense of groundedness in the physical past and a repository of common sense. Both types are also prone to giving proverbs and these may become stock sayings, also something I've found in the Type 9 description.
So yes, that is my correlation. I do realize that one's Enneagram type is assessed in a different manner than CT and also includes wings and sx/sp/so variants, so of course it's not going to correlate perfectly, and one also has to look at development levels to add some layers.