I find the definition of a right rigid object here https://en.wikipedia.org/wiki/Rigid_category
But was is the definition of a left rigid object? it says that it is a similar definition. But similar in what way?
Is it an object for which there exists an object $X^*$ and maps $\text{ev}:X\otimes X^*\rightarrow I$ and $\text{coev}:I\rightarrow X\otimes X^*$ such that some diagram commutes?
Er, I think you're mixing up left and right: the Wikipedia page defines left rigid objects, not right rigid objects.
To define a right rigid objects, you just reverse the order of all the products. In other words, a right rigid object is a left rigid object in the monoidal category obtained by reversing the order of the monoidal product. Concretely, this means $X$ is right rigid if there exists an object $X^*$ and maps $X\otimes X^*\to I$ and $I\to X^* \otimes X$ for which the compositions $$X\to X\otimes (X^*\otimes X)\to (X\otimes X^*)\otimes X\to X$$ and $$X^*\to (X^*\otimes X)\otimes X^*\to X^*\otimes (X\otimes X^*)\to X^*$$ are both the identity.