My Answer: The soul is a “part of the Creator from Above.” No person born in our world has it from birth. The only people who have a soul are those who have attained at least the minimal quality of bestowal, the quality of the Creator.
With the case you are asking about, a person corrected himself, attained the quality of bestowal, built his soul (vessel of perception – Kli), and suddenly this vessel acquired a specific form and was filled with the Light called “the soul of the Ari.” The soul is the corrected desire (this is “the body” of the soul), and the Light that fills it (the Light of the soul) is the revelation, the Creator, which clothes into you. That is why the soul is called a “part of the Creator from Above.”
Receiving the soul of the Ari means thinking like him, feeling like him, being instead of him, seeing through his eyes, and feeling with his senses just like him. Baal HaSulam received the soul of Ari, which is the same soul that was passed from Adam to Abraham, from Abraham to Moses, from Moses to Rashbi, from Rashbi to the Ari, and from the Ari to Baal HaSulam. This is a special soul through which the Creator reveals the method of merging with Him during the generations when a special spiritual change has to occur. If the Creator sent us a teacher like Baal HaSulam, who had the soul of the Ari, this indicates that we are living in the generation of the last, general correction.
The soul constantly changes and reincarnates. Throughout the course of a Kabbalist’s lifetime, he always has a different Partzuf and a different degree. A person’s corrected qualities that have the intention of bestowal, in which the Light can be clothed, is called a soul. The Kli is called the body of the soul, and the Light is called the soul. This is the Light of NaRaNHY, but in general it is called “Neshama” (soul) because until we reach the complete correction, we can only correct this degree completely.