Skill Up & Ascension MaterialProof that you've been acknowledged as a hero. Many possess it, but they aren't conscious of it so they forget.
Okeanos: Pirate Ship (Free Quest) is the best AP-saving & time-saving location to farm Hero's Proof (Proof of Hero). You'll get around 6x Hero's Proof by spending 120AP.
Here is a list of servants that need Hero's Proof to perform Ascension or Skill-Up.
Rashoumon event (Ibaraki Douji event)
Murder at the Kogetsukan
Inheritance of Glory
5 Star Servants Tier List: Best Servants [5/9 updated!]
Best Servants [★4]: 4 Star Tier List
FGO Reroll Tier List [5/11 updated!]
Mysterious Heroine X (Alter)
Victor from the Moon
Octuplet Crystals (Octuplet Twin Crystals)
Marie Antoinette (Rider)
Cu Chulainn (Lancer)
Gaius Julius Caesar
Piece of Year 2030 (A Fragment of 2030)
Dumplings Over Flowers