Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,

$A^\ast=C,$

$B^\ast=C,$

can we state that $A=B$? Is the dual cone of a cone is unique?

the definition of dual cone here is:

The dual cone C* of a subset C in a linear space X, e.g. Euclidean space $R^n$, with topological dual space X* is the set

$C^* = \left \{y\in X^*: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \}$,