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We prove that a large class of natural transformations (consisting roughly ofthose constructed via composition from the "functorial" or "base change"transformations) between two functors of the form $\cdots f^* g_* \cdots$actually has only one element, and thus that any diagram of such mapsnecessarily commutes. We identify the precise axioms defining what we call a"geofibered category" that ensure that such a coherence theorem exists. Ourresults apply to all the usual sheaf-theoretic contexts of algebraic geometry.The analogous result that would include any other of the six functors remainsunknown.

Obvious natural morphisms of sheaves are unique