category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
Given an adjoint pair of functors $(f^\ast \dashv f_\ast)$ or $(f_! \dashv f^\ast)$ between two monoidal categories such that $f^\ast$ is a strong monoidal functor, then the projection morphism is the canonical natural transformation of the form
or
respectively. If these morphisms are equivalences then one often calls them the projection formula or the reciprocity relation.
Examples include the six operations setup in Grothendieck context and Wirthmüller context, respectively (in a transfer context both pushforward maps satisfy their projection formula). In particular in representation theory and in formal logic reciprocity is also called Frobenius reciprocity, see there for more.
For more examples see also at MO Where do all the projection formulas come from?
Discussion in real cohomology for integration of differential forms:
A general abstract account is in
For the Grothendieck context of quasicoherent sheaves in E-infinity geometry the projection formula appears as remark 1.3.14 in
See also
MathOverflow, Where do all the projection formulas come from?
MathOverflow, Ubiquity of the push-pull formula