Let $(M,\xi)$ be a transversally orientable contact manifold, that is, there exists a form $\alpha \in \Omega^1(M)$ such that $\xi = \ker \alpha$. Then we can associate to $(M,\xi)$ its symplectisation $(\mathbb{R} \times M,d(e^t\alpha))$, a symplectic manifold. I wondered, if there is a categorical setting for this process. I mean, naively, we could consider symplectisation as a map on objects $$S \colon \mathsf{TOCont} \to \mathsf{Symp}$$ where $\mathsf{TOCont}$ denotes the category of transversally orientable contact manifolds as objects and maps $F \in C^\infty(M,\widetilde{M})$ such that there exists a nowhere vanishing function $f \in C^\infty(M)$ with $F^* \widetilde{\alpha} = f\alpha$ as morphisms $F \colon (M,\xi = \ker \alpha) \to (\widetilde{M},\widetilde{\xi} = \ker \widetilde{\alpha})$. Likewise, $\mathsf{Symp}$ denotes the category with objects symplectic manifolds and morphisms $F \colon (M,\omega) \to (\widetilde{M},\widetilde{\omega})$ such that $F \in C^\infty(M,\widetilde{M})$ with $F^*\widetilde{\omega} = \omega$.

Now the problem I am facing is the following: I would define $S$ on morphisms as follows. $$S(F) \colon (\mathbb{R} \times M,d(e^t\alpha)) \to (\mathbb{R} \times \widetilde{M},d(e^t\widetilde{\alpha}))$$ by $$S(F) := \operatorname{id}_{\mathbb{R}} \times F.$$ But then, if $F^* \widetilde{\alpha} = f\alpha$, we compute $$S(F)^* d(e^t\widetilde{\alpha}) = d(e^tf\alpha),$$ that is, $S(F)$ is not a morphism in $\mathsf{Symp}$. If $f > 0$, we could use the definition $$S(F)(t,x) := (t - \log(f(x)),F(x))$$ and things would work out fine. However, this would impose a restriction on orientation.

I think everything boils down to the fact that if $(M,\xi = \ker \alpha)$ is a contact manifold, then also $\xi = \ker f\alpha$ for every nowhere vanishing smooth function $f$. But I guess the symplectisations are not symplectomorphic in general in this case, that is, a single t.o. contact manifolds admits different non-symplectomorphic symplectisations. Is that right? Do you have any idea how to turn symplectisation into a functor between appropriate categories?

Let $(M,\xi)$ be a transversally orientable contact manifold, that is, there exists a form $\alpha \in \Omega^1(M)$ such that $\xi = \ker \alpha$. Then we can associate to $(M,\xi)$ its symplectisation $(\mathbb{R} \times M,d(e^t\alpha))$, a symplectic manifold. I wondered, if there is a categorical setting for this process. I mean, naively, we could consider symplectisation as a map on objects $$S \colon \mathsf{TOCont} \to \mathsf{Symp}$$ where $\mathsf{TOCont}$ denotes the category of transversally orientable contact manifolds as objects and maps $F \in C^\infty(M,\widetilde{M})$ such that there exists a nowhere vanishing function $f \in C^\infty(M)$ with $F^* \widetilde{\alpha} = f\alpha$ as morphisms $F \colon (M,\xi = \ker \alpha) \to (\widetilde{M},\widetilde{\xi} = \ker \widetilde{\alpha})$. Likewise, $\mathsf{Symp}$ denotes the category with objects symplectic manifolds and morphisms $F \colon (M,\omega) \to (\widetilde{M},\widetilde{\omega})$ such that $F \in C^\infty(M,\widetilde{M})$ with $F^*\widetilde{\omega} = \omega$.

Now the problem I am facing is the following: I would define $S$ on morphisms $$S(F) \colon (\mathbb{R} \times M,d(e^t\alpha)) \to (\mathbb{R} \times \widetilde{M},d(e^t\widetilde{\alpha}))$$ by $$S(F) := \operatorname{id}_{\mathbb{R}} \times F.$$ But then, if $F^* \widetilde{\alpha} = f\alpha$, we compute $$S(F)^* d(e^t\widetilde{\alpha}) = d(e^tf\alpha),$$ that is, $S(F)$ is not a morphism in $\mathsf{Symp}$. If $f > 0$, we could use the definition $$S(F)(t,x) := (t - \log(f(x)),F(x))$$ and things would work out fine. However, this would impose a restriction on orientation.

I think everything boils down to the fact that if $(M,\xi = \ker \alpha)$ is a contact manifold, then also $\xi = \ker f\alpha$ for every nowhere vanishing smooth function $f$. But I guess the symplectisations are not symplectomorphic in general in this case, that is, a single t.o. contact manifolds admits different non-symplectomorphic symplectisations. Is that right? Do you have any idea how to turn symplectisation into a functor between appropriate categories?