The Categorification of a Symmetric Operad is Independent of Signature

Given a symmetric operad $P$, and a signature (or generating sequence) $\Phi$ for $P$, we define a notion of the "categorification" (or "weakening") of $P$ with respect to $\Phi$. When $P$ is the symmetric operad whose algebras are commutative monoids, with the standard signature, we recover the notion of symmetric monoidal categories. We then show that this categorification is independent (up to equivalence) of the choice of signature.