In Shinto, it is a premise that interpretations differ depending on the system. As one example, in Yamino Shinto, it is explained as follows:

Ōkan Uta-no-Tam (Great Cosmic Spirit)
→ (Movement of Nothingness) Lawful Activity
・ Law of cosmic creation, generation, and nurturing
・ Law of the four seasons, revolution, and rotation
→ (Activity of Somethingness) Phenomenal Activity
・ Manifestation of the sun, moon, and stars
・ Manifestation of all things
"Kigi Hiden P50"

・Ara-no-Tam (Rough Spirit): Courage, construction, destruction. (Physical) life. Rational energy.
・Nigi-no-Tam (Harmonious Spirit): Control, propulsion. Blood, internal organs.
・Sachi-no-Tam (Blessed Spirit): Harmony, construction, compassion. Emotional energy. Nervous system.
・Kushi-no-Tam (Strange Spirit): Justice, hope, thought. Bones and muscles.
"Gendō no Gendai-teki Kai-i (Modern Interpretation of Shinto) by Yamino Kioo, P137"

・Environmental aspects are related to Ara-no-Tam.
・Physical aspects are related to Nigi-no-Tam.
・Psychological aspects are related to Sachi-no-Tam.
・Spiritual aspects are related to Kushi-no-Tam.
"Gendō no Gendai-teki Kai-i (Modern Interpretation of Shinto) by Yamino Kioo, P272"

Naohi (One Spirit) is divided into two movements:
→ Movement of Nothingness
→ Kushi-no-Tam: The function of life, the source of life.
→ Sachi-no-Tam: Harmony and change, emotions, beauty.
→ Movement of Somethingness
→ Nigi-no-Tam: Energy, internal organs, functions related to hormones. Also corresponds to what is called a ghost (Nurumi-no-Tam).
→ Ara-no-Tam: Energy for the external actions of the body. Nerves, muscles, bones, etc.
"Gendō no Gendai-teki Kai-i (Modern Interpretation of Shinto) by Yamino Kioo, P386"

These can also be seen as similar to the five Koshas of yoga, but they also have a mental aspect, and it is likely that both aspects exist. The classification of the four spirits from a mental perspective is as follows:

・Ara-no-Tam: A rigid and hardened mental function, a revolutionary consciousness, a destructive consciousness, intuition, and expressive desire.
・Nigi-no-Tam: A flexible mind. A lazy mind that seeks stagnation. Emotional feelings. A desire for propulsion.
・Sachi-no-Tam: Perception, desire, love, awareness of harmony, and emotions that seek exploration.
・Kushi-no-Tam: A mind that seeks mystery, a creative consciousness of thought and contemplation, a calm and comprehensive intuition, and a driving force.
"Kigi Hiden P54"

The same author's "Shinto Nyūmon 2" (P225) includes a diagram, and the order from the aspect closest to the body is as follows:

・荒魂 (aramitama)
・和魂 (nigimitama)
・幸魂 (sachimitama)
・奇魂 (kusimitama)
・直日霊 (naohinomitama)

I believe the same author may have published a correspondence (interpretation) between these and the five koshas (sheaths) of yoga in another book, but I could not confirm this description in the author's book that I have now.

As a reference, a quote from a paper by a different professor is published on page 128 of the same book.

・荒魂 (aramitama): Constitutes the physical body.
・和魂 (nigimitama): Etheric body, astral body.
These two constitute the living physical body.

・幸魂 (sachimitama) and 奇魂 (kusimitama) are the consciousness body (causal body, karmic body).
・奇魂 (kusimitama) and 直日霊 (naohinomitama) are the spirit.

According to this, 奇魂 (kusimitama) appears twice.

Since it is a quote from a paper, it seems that the correspondence between the four souls of Shinto and the five koshas of yoga is not a common interpretation but rather the author's personal interpretation. In the first place, the interpretation of the four souls varies depending on the Shinto sect, so there may not be a common interpretation.

I will add a personal interpretation to each explanation (of course, this is not a common interpretation).

・荒魂 (aramitama)
　Yoga's Annamaya Kosha (food sheath)
・和魂 (nigimitama)
　Yoga's Pranamaya Kosha (vital energy sheath)
・幸魂 (sachimitama)
　Yoga's Manomaya Kosha (mind sheath)
　Yoga's Vijnanamaya Kosha (intellect sheath)
・奇魂 (kusimitama)
　Yoga's Anandamaya Kosha (bliss sheath)
・直日霊 (naohinomitama)
　Yoga (or Vedanta's) Atman (true self)

If you simply assign the five koshas to each, it does not make sense, so based on a certain degree of similarity, the correspondence is as follows. Of course, the definitions are different because the sects are different, but if you roughly consider them to be quite similar, it may be easier to grasp the overall picture.

It seems that there are established theories for this kind of story, but the details are different even within the same sect, and it is quite difficult to interpret.

One of the reasons for this is that the four souls not only classify things based on the material grossness and fineness, but also contain qualitative differences. The classification of grossness and fineness as shown in the figure above is certainly correct, but in addition to that, the fact that the qualities are associated with each from a spiritual perspective makes it difficult to understand because the concepts of quality and grossness/fineness are mixed.

As mentioned, there seems to be a general consensus on which elements correspond to each other, but regarding the classification of "spirit" and "matter," it seems to be clearly defined as follows:

Purusha corresponds to "naohimimata," and Prakriti corresponds to the "yonrei." "Yoga and Shinto," p. 363.

Purusha is the pure spirit in the Samkhya school of Yoga, and what is opposite to it is Prakriti, which is pure matter. Therefore, Purusha is also translated as "divine spirit," but in the Sanskrit meaning, it means "human being." Both spirit (divine spirit) and matter are always together, and it seems that Yoga and Shinto have similar ideological systems.

Furthermore, the same author describes the correspondence with modern spiritualism, theosophy, or spiritual terminology.

・Ara-mitama (rough spirit)
Physical body, etheric body
・Nigi-mitama (harmonious spirit)
Astral body, mental body. Ghost body. Semi-material spiritual body
・Sachi-mitama (blessed spirit) and Kushi-mitama (strange spirit)
Causal body, spiritual body. Conscious spiritual body. Does not belong to the realm of matter.
From "Yoga and Shinto," p. 516.

Whether the causal body (karana) is considered matter or not varies slightly depending on the school, but Honzan Hiraoka considers the karana to be matter, while here it is considered not to be matter. Since Prakriti is matter in Yoga, if Prakriti corresponds to the yonrei, then Sachi-rei and Kushi-rei should also be matter, but this seems to be a subtle and unclear area. I interpret the causal body according to Honzan Hiraoka's interpretation as matter.

By looking at the five koshas of Yoga, the Purusha and Prakriti of the Samkhya school in the Yoga Sutras, and the correspondence between spirituality and the yonrei, it can be seen that the ideology of the yonrei in Shinto is not so different and has many common points with them.