Law, Doflamingo and Vegapunk
Alright, so everyone should know about how during Law's confrontation with Joker, he mentioned "him" and what "he" worked his whole life for. Now, after some thinking, I think this man that Law is referring to is Vegapunk.
I'll start off my theory first by talking about Vegapunk. As we all know, Vegapunk is a scientific genius and a benevolent man. His benevolence is evident by how when he was young, he wanted to create something that would heat up his home island for his inhabitants. He also refused to experiment on children, showing he had good morals and was a good person.
I believe Vegapunk wanted to use his genius to help people. Specifically, he wanted to enhance the bodies of humans so their lives could be easier and to help humans with disabilities. This is how he has discovered how Devil Fruits act on humans, created the Gigantification experiments and how to design Pacifistas, he was studying the human body to design ways to enhance it. Vegapunk has shown to have a liking towards creating animals, as he created the artificial dragons on Punk Hazard and the artificial Devil Fruit that Momo ate, as well as artificial animals to be the labor force behind his failed heating mechanism for his home island.
Vegapunk is noted to be an old geezer, and to have started working in science since he was young. Bear that in mind.
Now, how does Law tie in to Vegapunk? Well, I believe Vegapunk and Law were once associated to each other. How you ask? Well, Law is from North Blue. he North is usually associated with cold weather. Almost all of his clothing have been some kind of winter clothes, as he constantly wears hoodies, sweatshirts or jackets during the sereis, as well as his hat, which appears to be made of fur and is probably warm. Now, where did Vegapunk come from? A winter island. I believe this island is in North Blue and is where Law grew up in. Now here it mostly becomes speculation and ideas rather than solid facts. On this island, after a while, Vegapunk established some kind of school that taught science. After all, he's growing old, he doesn't want his work to die with him. Law went to this school ,or institution and learned how to be a surgeon here. Now you're probably wondering "Vega worked with tech, what does he know on medical science?" A lot actually, he was able to design artificial humans and animals, and figured out Devil Fruits affect the body, meaning he would had to have experimented on human bodies and that requires medical knowledge. I also believe Law got his hat from Vegapunk and looked up to Vegapunk. It was through his association with Vegapunk that Law knew about Punk Hazard, after all, it's supposed to be an island that hardly anyone knows exist, how would a rookie like Law know where it is? Well, there's another factor. Doflamingo.
At some point in the past, Doflamingo came to Vegapunk's island after hearing through the Underworld rumors that Vegapunk had created an artificial DF. Seeing the profit and power this could give him, Joker proposed Vega supply him with the Fruits, or SMILES as they will become known as, and Joker would sell them, being a broker. However, Vega refused, as he intended for SMILES to be used to help people, not to be sold as weapons. Disappointed, Joker turned to Vega's underling, Caesar and that's where the two formed their alliance. It was around this time Joker met Law and that's how the two became known to each other. Law then joined the Don Quixote Pirates, and we know how that ended.
You're probably wondering why I believe Law looks up to Vegapunk. Well, it's because they have some subtle similarities. If my theory is right, Vega created Zoan fruit to help people. When Law came to Punk Hazard, he helped the people there by replacing their lost limbs with what? ANIMAL PARTS. Law didn't have to help them, but was inspired by Vega's kind ways to do so. Also, like Vegapunk, he is appalled to the use of children in experiments.
Now fast forward to the post-war arc. Law mentioned that he didn't want to rush into the New World. I believe this was because he caught wind of hearing about artificial DF's about to be sold under the production of a secret scientist and how he learned about SAD and stayed in the Grand Line so he could gather more information before moving into the New World. During the timeskip, he fully learned of SAD and how it was being used for weapons. This angered him because Law knew Vega wouldn't want this. Grateful to Vega for supplying Law's desire of being a surgeon(Remember how Oda depicted Law looking like a surgeon with the frog and scalpel since he was a kid?), Law has now set out to set things right for Vega, who is unable to do so since he is old and appears to be a regular human who is just a genius.
So, in short, the "him" Law is referring to is Vegapunk, the work that "he" spent his entire life working on is helping humans, and the event 13 years ago could be that maybe Joker sabotaged one of Vega's workings or turned Caesar to his side.
Anyway, I know it's far-fetched and rather baseless at times, but it's an idea I had and wanted to share with you guys.