Let $Z$ and $X$ be smooth manifolds. For simplicity, let them be oriented. Denote by $\mathscr D^*(X)$ the space of currents on $X$, i.e. the topological dual to the space $\Omega_c^*(X)$ of compactly supported smooth differential forms on $X$. Is the following true:
Assigning to $K\in \mathscr D^*(Z\times X)$ the function $$ \mathscr K\colon \Omega_c^*(Z)\to \mathscr D^*(X) $$ given by $$ (\mathscr K\sigma)(\tau)=\mathscr K(\sigma\otimes \tau) $$ is a bijection onto the continuous linear maps $\Omega_c^*(Z)\to \mathscr D^*(X)$.
This is of course a version of the Schwartz kernel theorem, but one I have failed to find a reference for.