Denote by $\mathcal{D}'_X$ the sheaf of distributions on a smooth manifold $X$.
Let $M$ and $N$ be smooth manifolds and $\Phi: M \to N$ a submersion. Then $\Phi$ defines a unique morphism of sheaves $\Phi^*: \mathcal{D}'_N \to \Phi_*\mathcal{D}'_M$ which satisfies $\Phi^*_U f = f\circ \Phi|_U$ for $f \in C(U), \ U\subseteq M$ open.
Now, if $\Phi$ is surjective then $\Phi^*$ should be an injective morphism of sheaves.
Is this true and how can one prove this? What would be a counterexample if it does not hold?
On the one hand it should be true since it holds for maps between sets that if $g:X \to Y$ is surjective and $f\circ g = f' \circ g$ where $f,f':Y \to Z$ then $f = f'$.
On the other hand I don't think that one could reduce the above problem to the case to show injectivity of $\Phi^*: \mathcal{D}'(V) \to \mathcal{D}'(U)$ where $U \subseteq \mathbb{R}^m$ and $V \subseteq \mathbb{R}^n$ are open subsets and $\Phi: U \to V$, because the preimage $\Phi^{-1}(W)$ of coordinate neighborhoods $W$ might not be coordinate neighborhoods (even if $W$ is made smaller and smaller). But even in this case I don't see a way to prove this. I think a density argument with density of test functions $\mathcal{D}(U)\subseteq \mathcal{D}'(U)$ doesn't work.
So, I doubt that it is true but still think that it would be natural if it was true. (Also https://math.stackexchange.com/q/4986397)
