  We prove that the proper amalgam of matroids $M_1$ and $M_2$ along their
common restriction $N$ exists if and only if the tropical fibre product of
Bergman fans ${B(M_1) \times_{B(N)} B(M_2)}$ is positive. We introduce tropical
correspondences between Bergman fans as tropical subcycles in their product,
similar to correspondences in algebraic geometry, and define a "graph
correspondence" of the map of lattices. We prove that graph construction is a
functor for the "covering" maps of lattices, exploiting a generalization of
Bergman fan which we call a "Flag fan".
