  Quantum $L_\infty$ algebras are higher loop generalizations of cyclic
$L_\infty$ algebras. Motivated by the problem of defining morphisms between
such algebras, we construct a linear category of $(-1)$-shifted symplectic
vector spaces and distributional half-densities, originally proposed by
\v{S}evera. Morphisms in this category can be given both by formal
half-densities and Lagrangian relations; we prove that the composition of such
morphisms recovers the construction of homotopy transfer of quantum $L_\infty$
algebras. Finally, using this category, we propose a new notion of a relation
between quantum $L_\infty$ algebras.
