Remember, Kawhi -- who was under contract with Nike from 2011 to 2018 -- claims he got a cease and desist in late '18 from Swoosh execs who claim they own the copyright to the "Claw Design" and forbid the NBA star from using it elsewhere.

In his lawsuit, filed in June 2019, Kawhi claimed he came up with the design while sketching around in college back in 2011. His version consists of his initials K.L. and jersey number #2 in a hand design. Leonard claims since he created the logo, he should have the rights to it.

But, Nike says that's simply not the case ... explaining they collaborated with Kawhi in 2014 to come up with an original design. Nike claims they submitted several concepts to him with a "number of proposed designs."

In new court docs obtained by TMZ Sports, Nike admits Kawhi had sent them a copy of his K.L. hand-design sketch as part of the collaborative process.

But, Nike says they eventually settled on an "original" version that is different from Kawhi's sketch.

Nike claims Kawhi signed off on the company using the final "Claw Design" in 2014 -- so they filed a trademark application to effectively own the rights to the image.

Nike says the company found out in 2017 Kawhi also filed an application to trademark the image -- and they were pissed ... saying he has no rights to it and had "fraudulently claimed to be the author and sole owner of the Claw Design."

In their suit, Nike says Kawhi even acknowledged publicly that Nike should be given "all the credit" for the design for turning a "rough draft" into a professional logo.

Nike is countersuing for copyright infringement, fraud and breach of contract -- and they're looking for a judge to stop Kawhi from using the design. They also want unspecified monetary damages.