What would be the dual of the statement "For categories A,B and functors S,T: A->B, X is a natural transformation X: S->T"?
I understand that the dual of "S is a functor from A to B" is itself, but for a natural transformation where you actually assign arrows Sa->Ta for every object a in A, would the dual statement be "X is a natural transformation T->S"? More precisely, what are the atomic statements that make up "X is a natural transformation S->T"?
The dual statement is that $X^{op}$ is a natural transformation $T^{op} \to S^{op}$, where $S^{op} : A^{op} \to B^{op}$ is defined "like $S$", the same for $T^{op}$, and $X^{op}$ is defined "like $X$" too.