1-morphisms and 2-morphisms of $\mathcal{S}$ which are invertible with respect to the tensor product, i.e. if it takes values in the Picard sub-2-groupoid of $\mathcal{S}$.

In this article we use the 2-category of Kapranov–Voevodsky (KV) 2-vector spaces ${{2\mathbf{Vect}_{\mathbb{C}}}}$ [@KV] as codomain of our field theories. The codomain will also be referred to as the *coefficients*. For a discussion of alternative targets see [@BDSPV15 Appendix A]. The most common coefficients for extended field theories contain ${{2\mathbf{Vect}_{\mathbb{C}}}}$ as a sub-2-category. The objects of ${{2\mathbf{Vect}_{\mathbb{C}}}}$ are KV 2-vector spaces, i.e. semi-simple abelian ${{\mathbf{FinVect}_{\mathbb{C}}}}$-module categories with a finite number of isomorphism classes of simple objects. We denote the action of a finite-dimensional complex vector space $V$ on a 2-vector space $\mathcal{X}$ by $V*\cdot \colon \mathcal{X}\rightarrow \mathcal{X}$. The 1-morphisms are linear functors, i.e. ${{\mathbf{FinVect}_{\mathbb{C}}}}$-module functors, and the 2-morphisms are natural transformations. The Deligne product $\boxtimes$ makes ${{2\mathbf{Vect}_{\mathbb{C}}}}$ into a symmetric monoidal bicategory. Note that every KV 2-vector space is equivalent to $({{\mathbf{FinVect}_{\mathbb{C}}}})^n$ for some integer $n \ge 0$. Using this equivalence we can express a linear functor up to natural isomorphism as a matrix with vector spaces as entries. A natural transformation can be described as a matrix with linear maps as entries [@Mor11].

Construction 