Call this as my way to pickup my Japanese lessons or another One Piece meta post but I find it fascinating that Doflamingo consistently describes Law's motivations for fighting him in Dressrosa as resentment and revenge. On the other hand, Law uses several terms to encapsulate why he was truly there.
Doflamingo says 逆恨み (sakaurami) which means unjustified resentment or unreasonable grudge. Law, in turn, denies this and says he is there to fulfill that person's 本懐 (hon'kai) which means (one's) long-cherished desire.
What is that desire, you may ask? Corazon shared it when he spoke for the first time with Law: His purpose/motivation/goal is to stop Doflamingo's 暴走 (bousou), which is translated as rampage.
(Later on, it was further revealed in the flashback and confirmed by Doflamingo that Corazon was trying to save Dressrosa--and the ever manipulative little shit hoped to rattle Law's feelings about screwing up in passing the message that Corazon asked him to do 13 years ago but he was able to keep his calm.)
The next time that Law thought about his motivation, specifically, the moment that he will be truly free is the time that Corazon's 思い (omoi) will be fulfilled/granted/come to fruition. 思い (omoi) could be translated differently depending on the context. For this one, it refers to desire or wish--a nod to Corazon's purpose, to stop Doflamingo.
Doflamingo didn't even scrimp with his descriptions as he used both 恨み (urami) which means resentment, grudge, malice, bitterness; and 復讐 (fukushuu) directly translated as revenge. He even says that Law is acting like Corazon's vengeful ghost. He views Law's persistent defiance to carry out Corazon's wish as pointless, even going through the effort of saying that Law has been overtaken with his feelings--however, when he saw the Corazon at the back of Law's coat, he riddled him with bullets and even saying that both Law and Corazon's curse have ended.
Lastly, and I think this is where Law's motivation reached its reveal in full circle. He says that he lived for the past 13 years to do this and he used 討つ (utsu) which, in this context, can be translated as to attack, to destroy, to defeat, and surprisingly, to avenge. This part is up for interpretation, really.
(My two cents on it is Law's main focus on wanting to take down Doflamingo stemmed from wanting to carry out Corazon's purpose, however, I am not necessarily dismissing the part of wanting to take responsibility and avenge Cora's death.)
Ending this self-indulgent meta post with Law giving the middle finger because Toei decided to not include this badass moment. LMAO.