  In a triangulated symmetric monoidal closed category, there are natural
dualities induced by the internal Hom. Given a monoidal functor f^* between two
such catgories and adjoint couples (f^*,f_*) and (f_*,f^!), we prove the
necessary commutative diagrams for f^* and f_* to respect certain dualities,
for a projection formula to hold between them (as duality preserving functors)
and for classical base change and composition formulas to hold when such
duality preserving functors are composed. This framework is for example useful
to define push-forwards for Witt groups.
