$\def\duevec{2{\rm Vect}}$See here for the notation.

The 2-category of 2-vector spaces is extremely rich in structure: I'm interested in studying some of its properties in the following perspective.

**Fact 1**: there is an involution of $\duevec$ transposing 1-cells and applying transposition entry-wise to 2-cells.

**Fact 2**: there is a semi-tautological embedding ${\rm Vect}\hookrightarrow \duevec$ which is constant on $\langle 1\rangle$ on 0-cells, and sends $V$ into the endomorphism of $\langle 1\rangle$ determined by the integer $\dim_KV$.

Now.

Let $\cal K$ be a 2-category which is monoidal closed, and endowed with an "involution" (for the purpose of this discussion, an involution is a functor doing the same job that $(-)^{op}$ does on $\duevec$ in **Fact 1** above). Let's call an object $\Omega$ of such a monoidal closed 2-category **lunatic** if it has the following properties:

- the 1-cell $\phi^* = [ \phi^{op},\Omega]$ has a left adjoint for each $\phi : X\to Y$; denote $\phi_! \dashv \phi^*$ this adjunction.
- there is a map $X \to [X^{op},\Omega]$ which is the unit of a monad structure for the 2-functor $T=[\,\_\,^{op},\Omega]$
- the multiplication of this monad is obtained applying $T$ to $\eta_X$, so that $\mu_X = T\eta_X : [[X^{op},\Omega]^{op},\Omega]\to [X^{op},\Omega]$

Question (in case the question here has affirmative answer): does $2{\rm Vect}$ has a lunatic object? Is such an object the ground field under the embedding ${\rm Vect}\hookrightarrow 2\rm Vect$?