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 $... f^* g_* ...$ 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.
Keywords: commutative diagrams, coherence theorem, string diagrams, pullback, pushforward
2010 MSC: Primary 14A15; Secondary 18D30, 18A25
Theory and Applications of Categories, Vol. 29, 2014, No. 4, pp 48-99.