  This paper is the second in a series of five that together prove the
geometric Langlands conjecture. Our goals are two-fold:
  (1) Formulate and prove the Fundamental Local Equivalence (FLE) at the
critical level;
  (2) Study the interaction between Kac-Moody localization and the global
geometric Langlands functor of ref. [GLC1].
  This paper contains an extensive Appendix, whose primary goals are:
  (a) Development the theory of ind-coherent sheaves in infinite type;
  (b)Development of the formalism of factorization categories.
