In a triangulated closed symmetric monoidal category, there are natural dualities induced by the internal Hom. Given a monoidal exact functor $f^*$ between two such categories and adjoint couples $(f^*,f_*)$, $(f_*,f^!)$, we establish the commutative diagrams necessary for $f^*$ and $f_*$ to respect certain dualities, for a projection formula to hold between them (as duality preserving exact functors) and for classical base change and composition formulas to hold when such duality preserving functors are composed. This framework allows us to define push-forwards for Witt groups, for example.
Keywords: closed monoidal category, commutative diagram, duality, Witt group
2000 MSC: 18D10
Theory and Applications of Categories,
Vol. 22, 2009,
No. 6, pp 136-198.