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In his book 'A Beautiful Question', physicist Frank Wilczek arguesthat symmetry is 'nature's deep design,' governing the behavior of theuniverse, from the smallest particles to the largest structures. Whilesymmetry is a cornerstone of physics, it has yet to be found tohave widespread applicability to describe biological systems. In thiscontext, we study biological graphs and explore the relationshipbetween the structural network and the emergent clustersynchronization of regions of interest (the functional network). Weexplain this relationship through a different kind of symmetry thanphysical group symmetry, derived from the categorical notion ofGrothendieck fibrations. This introduces a new understanding ofbiological networks, including brain networks, by proposing a localsymmetry theory that accounts for how the structure of the brainnetwork determines its coherent activity. Our findings suggest thatthe network symmetry at the local level determines itsfunction, and we can understand this relationship from theoreticalprinciples.