In the last chapter, we saw that 1 year after the fight with Sakazuki, Kuzan was on an island, met the Blackbeard pirates, and he was asked to join the crew. Now in the present, he is fighting against Garp to prevent Koby from escaping the island.
Let’s start by saying that how Kuzan acted with Teach sounded fishy to me; he freezed his comrades and waited for Blackbeard in a tavern, then gained their trust. Talking about his fight, and he seemed to know nothing about the man marked by flames. Then he started talking about Jaguar D. Saul that rebelled against the Government and how he had to take his life in his own hands after seeing a little girl cry.
One Piece is a manga that talks about dreams, ideals and inherited wills, and being Garp former prodigy, Kuzan might have also inherited part of his ideals, as Garp remind him in the last chapter “Live in the Present!”.
I think that after Marineford/Fight with Akainu, knowing that the world was changing and after seeing the Gura Gura in the hands of Blackbeard, Kuzan might have thought that was the time to make his move and create a new way to solve things with pirates.
We know that the Sword is an organization that can act on free will, independently from the World government and marines, and we know they use spies as X Drake, they pursue justice in their own way; but we don’t know if there is a hierarchy or a leader.
My speculation is that Kuzan founded the Sword, but not in a public way, but he spread his ideals and pushed some marines to create the concept of Sword on their owns, being the secret founder of the organization.
After that he went undercover to infiltrate the Blackbeard pirates and watch them from the inside, ready to act at the appropriate moment, because he realized that Teach is the real threat to this World.
Kuzan will remain loyal to his job until the end (reminds me Vegapunk saying to Dragon “remember our mission”) and he will give his all in the battle against Garp without mercy.