Using the theory of non-commutative geometry in a braided monoidal category, we improve upon a previous construction of non-commutative families of instantons of arbitrary charge on the deformed sphere Sθ4. We formulate a notion of non-commutative parameter spaces for families of instantons and we explore what it means for such families to be gauge equivalent, as well as show- ing how to remove gauge parameters using a non-commutative quotient construction. Although the parameter spaces are a priori non-commutative, we show that one may always recover a classical parameter space by making an appropriate choice of gauge transformation.
