  We prove that a large class of natural transformations (consisting roughly of
those 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 maps
necessarily commutes. We identify the precise axioms defining what we call a
"geofibered category" that ensure that such a coherence theorem exists. Our
results apply to all the usual sheaf-theoretic contexts of algebraic geometry.
The analogous result that would include any other of the six functors remains
unknown.
