This functor is positivity-preserving. By the way, you might enjoy thinking about what happens if you iterate second quantization. Now, an interesting object in physics is Minkowski space. We can also express the principal of general covariance and the principal of gauge-invariance most precisely by saying that observables are functorial. The point is, that a category is really a generalization of a group.
Group representations are really only a special case of category representations.
This idea was sold to me by Minhyong Kim. But then I wound up needing category representations in my own work on quantum gravity. A representation of a group, if we think of a group as a category as Sibley suggests, is just a functor from that category to the category Vect of vector spaces. So we can define a representation of a category to be a functor from that category to the category of vector spaces. Tangles are like braids but the strands can double back on themselves and there can also be closed loops.