In 2014, through TBA considerations on N=2 SUYM in four dimensions, Gaiotto conjectured a Lagrangian correspondence between holomorphic Lagrangians in the Dolbeault moduli space of Higgs bundles and the de Rham moduli space of holomorphic connections. His original conjecture was formulated for the Lagrangian of opers. The conjecture was solved in 2016 for holomorphic opers in my paper with Dumitrescu, Fredrickson, Kydonakis, Mazzeo, and Neitzke. Then Collier and Wentworth, using our analysis method, extended the correspondence for more general Lagrangians consisting of stable points. In my talk, I will present an algebraic geometry description of the Lagrangian correspondence of Gaiotto, based on the work of Simpson using VHS and Deligne connections.