Jungs idea of development of the self or individuation(also referred as enlightenment in many systems of belief) is closely related to the process of enlightenment in alchemy, Jung was an alchemist and much of his theories are derived from it. In alchemy there are seven steps to this process, which i will go through on this topic. Even tho this process of individuation is much larger than just the differentiation of the functions, i think that in order to differentiate the functions, you are forced to differentiate other parts of the psyche aswell, because the undifferentiated functions are connected to the contents of the unconscious, such as the shadow and anima. Also it should be noted that when i(and Jung) talk of irrational forces of the unconscious, it is also referred as the rational function that is less developed, mainly because when it comes through the unconscious, it comes through intuitive perception(perception of the unconscious material), which is irrational.
Yes its quite a wall of text, so for all you lazy people i separated the parts in which i explain briefly how they relate to typology. In my opinion it is better to see the whole picture, which the explanations on typology doesent offer so well.
1. Calcination: The first step is the humbling of the ego. Because the ego works as the barrier between the conscious and the unconscious mind, it is necessary to question ones current view of the self and realize that even tho we need ego to process things consciously and to function, it also works as an obstacle on becoming a whole and in realization of the true self. This process will not be easy and usually in order for the ego to bow down in front of the forces of the unconscious, there must be some major dissatisfaction with current state of self or psychic crisis, such as depression. Jung thought that depression occurs when powers of the unconscious wants to get on the surface, but ego keeps blocking them and causing loss of mental energy due to all energy being directed on things that ego wont let to the surface, thus causing apathy etc. This step is where the ego bows down(to the irrational and so far unknown forces of the unconscious) and this process of calcination is symbolized as burning to ashes, which represents the pain which ego(/consciousness) feels in this process quite well.
In typology, this could be seen as letting go of the current mid set, which consists of the developed functions(mostly dom and aux to lesser extent) which are tied to the ego.
2. Dissolution: In alchemy which process is symbolized with dissolving the ashes to water. This is the further breaking down of the artificial structures of the psyche, by total immersion in the personal unconscious(complexes, which are the structures of this), shadow(which is the neglected parts of the self, such as inferior function and to lesser extend tert) and anima(which is the feminine side in mans psyche, also soul) or animus(masculine side in womans psyche, also spirit). This process is unconscious to a large extend, in which ego lets go of the control and allows buried material from the unconscious to come to surface. Opening the "flood gates" and generating new energies(new ways of thinking) from the water held back.
In typology, this is when the inferior(and possibly tert to some extend) comes to surface, generating new ways of functioning and is more likely than not giving us hard time.
3. Separation: In alchemy this is referred as to filtering of this solution created and discarding any ungenuine or unworthy material. It is the rediscovery of our essence and the reclaiming of material previously rejected by the rational part of our minds. This process is for most part conscious process in which we review our formerly rejected material and decide what to let go of, and what to reintegrate to our refined personality. Letting go of our self-inflicted restraints to our true nature, so that we can shine through.
In typological terms, this is where the inferior is reexamined and parts of it that were influenced by our unhealthy complexes are discarded and parts of it that are actually good for us and helpful are integrated to our personality. This is where the actual differentiation of the inferior happens.
4. Conjunction: In alchemy this is referred as recombining the separated elements to form a new substance. This is the process in which the different aspects of our selves earlier separated, are used to build a new more full personality. This is where the irrational forces of the unconscious and rational forces of the conscious can be combined to form a new whole, new belief system(of values about self) and intuitive state where the the powers of the unconscious can freely become conscious. This is referred as the lesser stone in alchemy, which helps the person to clearly see what has to be done to achieve lasting enlightenment, which is the ultimate union of the different parts of our self into a new lasting whole, which Jung referred as individuation.
In typological terms, this is the ability to consciously use all of the four functions of thinking, feeling, sensation and intuition, but their usage is still limited and might still be bothering to the person to some degree.
5. Fermentation: In alchemy this is seen as a two step process, which begins with the death of the hermaphroditic child from the conjunction and its resurrection into a new level of being. Hermaphrodite in alchemy is a symbol of a union of the rational and irrational forces in a person, so no, you wont actually grow both penis and vagina in the process . Fermentation process could be seen as a sort of secondary lesser crisis with the new self, in which new life is introduced into the product of conjunction, further strengthening the newly developed self and ensuring its survival. More pure material flows from the still bothered psyche, giving spiritual inspiration and meaningful visions about the new self.
In typology this could be referred as an difficulty of integrating the newly developed functions in real life situations.
6. Distillation: In alchemy this is referred as boiling down the fermented material to make it more pure. Turning the negative impulses of the unconscious into acceptable and productive use, ensures that none of the material of inflated ego are incorporated into the next and final process.
In typology this could be referred as learning to use the functions better in more situations and further refining the skills.
7. Coagulation: This is where the purified distillation turns solid. It is sensed as a new confidence which is beyond all things. It incarnates and releases the ultimate materia of the psyche, which is referred as the astral body in some way of thinking and is the philosophers stone in alchemy. Using the philosophers stone, alchemists believed that they could exist in all levels of reality. This is the enlightened state, which so many different practices from buddhism to christianity(although the whole idea is pretty twisted in christianity, in my opinion) aim for.
In typology this existing in all levels of reality naturally means to be conscious and freely use all of the four functions without the shadow, anima or our complexes interfering with the inferior or other functions.