Contents: Introduction: Synopsis of the paper Acknowledgements. 1. Notation. 2. Preliminaries on Lagrangian representations, proper sets and Optimal Transport. 2.1. Lagrangian representations. 2.2. Proper sets. 2.3. Optimal transport and duality. 3. Optimal transport and duality. 4. The global theory of forward untangling. 4.1. Subadditivity of untangling functional. 5. Uniqueness by forward untangling. 5.1. Decomposition and disintegration. 5.2. Composition rule. 6. Monotone vector fields. References.