adjoint of $\Psi_{E_a, E_b}$. Hence the adjoint $\Psi_{b,a}$ of a half-form morphism $\Psi_{a,b}$ is a half-form morphism. Furthermore, since the only automorphisms of a half-form bundle are ${\operatorname{id}}$ and $-{\operatorname{id}}$, one has $$\Psi_{b,a} {\circ_{{{\mathcal{D}}}}}\Psi_{a,b}
= \pm {\operatorname{id}}, \qquad \Psi_{a,b} {\circ_{{{\mathcal{D}}}}}\Psi_{b,a} = \pm {\operatorname{id}}.$$ Deforming from $E_a = E_b$ to remove the sign ambiguity, we obtain that $\Psi_{b,a}$ is the inverse of $\Psi_{a,b}$.

Symplectic linear category with half-forms {#sec:sympl-line-categ}
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We introduce now a category ${{\mathcal{D}}}$ with objects the quadruples $(S, E, \delta, \varphi)$ consisting of a symplectic vector space with a positive polarization and a half-form line. Let us define the morphisms.

Consider two symplectic vector spaces $S_a$ and $S_b$ with positive polarizations $E_a$ and $E_b$ respectively. A symplectic linear isomorphism $g$ from $S_a$ to $S_b$ sends isomorphically $E_{a}$ to the positive polarization $g E_{a}$ of $S_b$. So composing the pull-back by $g^{-1}$ with the morphism $\Psi_{ g E_a, E_b}$ from $\wedge^{{\operatorname{top}}} (gE_a)^*$ to $\wedge ^{{\operatorname{top}}} E_{b}^*$, we obtain an isomorphism $$\Psi_{g, E_a, E_b} := \Psi_{ g E_a, E_b} \circ \bigl( g^{-1} \bigr)^* : \wedge^{{\operatorname{top}}} E^*_{a}
\rightarrow \wedge^{{\operatorname{top}}} E^*_{b}.$$ The morphisms from $a= (S_a, E_a, \delta_a, \varphi_a)$ to $b= (S_b, E_b, \delta_b,
\varphi_b)$ are defined as the pairs $( g, 