It investigates weakly efficient solutions of constrained set optimization problems. By introducing the notions of C-convex-like and C-boundedness, and referencing the Lagrange multiplier approach commonly used in vector optimization, a Lagrange multiplier framework for weakly efficient solutions is developed in set optimization. This framework enables the transformation of constrained set optimization problems into equivalent unconstrained formulations. Moreover, it establishes the relationship between the solutions of the original and transformed problems and substantiates the theoretical results with numerical examples. The proposed Lagrange multiplier theorem offers an extendable theoretical foundation for further studies in set optimization.