I am learning the book of Ambrosio-Gigli-Savaré "Gradient flow". However, I am blocked at the proof of proposition 8.5.2 (Optimal displacement maps are tangent). I don't understand where $\nabla \phi_\epsilon$ come from and why absolute continuity of $\mu$ implies that $j_q(\nabla \phi_\epsilon)$ converge to $j_q(\nabla \phi)$.
If someone can give me some hints, I will appreciate it very much.