Let G be a groupoid. I'm wondering how to construct the free 2-group on G.

By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$ equipped with a functor $i:G\longrightarrow\mathcal{F}\left(G\right)$ such that for any 2-group $\mathcal{G}$ and any functor $F:G\longrightarrow\mathcal{G}$ there is a monoidal functor $F':\mathcal{F}\left(G\right)\longrightarrow\mathcal{G}$ and an isomorphism $\alpha:F'\circ i\Longrightarrow F$ such that the functor $F':\mathcal{F}\left(G\right)\longrightarrow\mathcal{G}$ is unique up to coherent isomorphism. Here is my attempt at a description of $\mathcal{F}\left(G\right)$ :

The objects are defined inductively by requiring that $\mathcal{F}\left(G\right)$ contain an object 1 , the objects of G and for each object $g\in G$ an object $\overline{g}\in G$ such that for any two objects $x,y\in\mathcal{F}\left(G\right)$ there is an object $x\otimes y$ in $\mathcal{F}\left(G\right)$ .

The 'prearrows' in $\mathcal{F}\left(G\right)$ are built as follows. We require that every arrow from G be a prearrow in $\mathcal{F}\left(G\right)$ . For each pair of prearrows $a:x\longrightarrow z$ and $b:y\longrightarrow w$ in $\mathcal{F}\left(G\right)$ there is a prearrow $a\otimes b:x\otimes y\longrightarrow z\otimes w$ . In addition we adjoin a prearrow $e_{g}:g\otimes\overline{g}\longrightarrow1$ for each $g\in G$ and we adjoin prearrows $\alpha_{x,y,z}:\left(x\otimes y\right)\otimes z\longrightarrow x\otimes\left(y\otimes z\right)$ , $\lambda_{x}:1\otimes x\longrightarrow x$ , and $\rho_{x}:x\otimes1\longrightarrow x$ for each $x,y,z\in\mathcal{F}\left(G\right)$ . The arrows are then given by equivalence classes of prearrows generated by the requirement that $\alpha,\rho,\lambda$ and e be isomorphisms, the naturality conditions on $\alpha,\rho$ and $\lambda$ , the condition that $\otimes$ is a functor, and the axioms of a monoidal category.

My question is, does this construction work? If not, can anybody give an indication of a construction that does? Also, to define the free symmetric 2-group on a groupoid, does it suffice to add prearrows $\gamma_{x,y}:x\otimes y\longrightarrow y\otimes x$ for each $x,y\in\mathcal{F}\left(G\right)$ and generate the equivalence relation from the axioms of a symmetric monoidal category?