Tosaka Tokiomi ('former' Archer camp) - Reached the "origin" [which can also be said as the "vortex of the source (of magic)"], that is, the "outside" of the world (the non-real world content). In fact, from its original intention, it is purely to complete Tohsaka's "family mission."

Webber Velvet (Rider camp) - simply proves his talent and worth by "winning the Holy Grail War and obtaining the Holy Grail." However, it can also be said that the intention of training oneself is mixed in.

Kenneth Elmloy Archibalud (Lancer camp) - to win (the detailed 'expression' wish is unknown); in fact (the 'inside' wish) in order to gain fame, Prove your excellence to the world (especially your fiancée). "As expected of a master and apprentice (the relationship with Weber is a teacher and student), the hopes in the heart are so similar..."

Sora Nazelle Sofiali (Lancer camp) - Originally not His personal purpose was to join the war just to help his fiancé (no real feelings); later he was attracted to Lancer (on purpose) and had a personal wish to "resurrect" Lancer and "be happily together" with Lancer.

Matou Kariya (Berserker camp)——(External:) Save Matou Sakura; (Back:) Live a happy life with Aoi, Rin, and Sakura, mother and daughter. "I have to point out that the "process" to achieve the goal is contrary to the goal "itself", so it is destined to end tragically. He is worthy of the title of "hard-working man" =　

Ame Ryuunosuke (Caster camp) - None. (He really didn’t know about the “Holy Grail War”, and Caster didn’t explain it clearly to him = =) If it really existed, it should be something like “obtaining research-level forbidden pleasures”.

Kotomine Kirei (Assassin camp) - Still in a period of confusion, he obeyed his father's orders and helped Tokiomi Tohsaka win for the benefit of the church.

(The 'post' Archer camp) pursues the reason why he was chosen by the Holy Grail. In other words, his purpose is to "want to know his purpose." "In fact, it can be explained as: wanting to understand the "self". 』

Emiya Kiritsugu (Saber camp) - eliminate all wars and promote world peace.

[Save the world? ] "As the saying goes - I want to be a partner of justice!" ~』

This is just a general explanation, no in-depth analysis, sorry~