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!]
Saint Quartz Farming Guide
Ryougi Shiki MAX Ascension/NP Guide
Best Servants [★4]: 4 Star Tier List
Kara no Kyoukai (The Garden of sinners) Event Rerun [JP]
Anastasia Nikolaevna Romanova
Device Transfer Guide
Oda Nobunaga (Berserker, Summer 2017)
Grand Puppet Master (Grand Puppeteer)
Lone Man's War