If you want to skip the side stories and watch only Luffy vs. Katakuri:

The fight between Luffy and Katakuri begins when Luffy, determined to protect his friends, confronts Katakuri, who had been tracking him. Luffy uses his Gear 4th form to take on Katakuri's incredible physical strength, enhanced by his Busoshoku Haki.

In the story, Luffy and Katakuri's confrontation takes place during the Whole Cake Island Arc, specifically in the chapter/episode 881-884 of the manga/anime series "One Piece".

The epic confrontation between Luffy and Katakuri marks a turning point in the story, demonstrating that even the strongest foes can be overcome with courage, friendship, and a strong sense of justice.

Here is a full breakdown of when and how the fight happens:

Moreover, Luffy's bravery and determination inspire his friends, and together, they work to rescue Sanji and defeat the Big Mom Pirates.

The Straw Hat Pirates, led by Luffy, infiltrate the Big Mom Pirates' territory, Whole Cake Island, to rescue Sanji, who was kidnapped by the Charlotte Family. Their goal is to enter the Mirimiri Maternity, where Sanji is being held.

The fight officially begins in Episode 850 , titled "A Ray of Hope! The Sunny Sets Sail!", and concludes in Episode 871 , "Finally, It’s Over - The Climax of the Intense Fight against Katakuri".

Let’s break it down by anime episode, manga chapter, and the overall arc.