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
Halloween Strike! Event Missions [JP Halloween 2017]
Best Craft Essence: Craft Essence Tier List
Cu Chulainn (Alter, Berserker)
Jeanne d’Arc (Alter)
Cheer for Master
Demon King of the Sixth Heaven
Talk on the Heated Sands