Fourier--Mukai transforms

Kernel functors between triangulated categories, stated for an abstract correspondence; the convolution of kernels as supplied data; kernel-presented adjoints of a transform, also as supplied data, together with the ledger that splits one such adjoint into the three constituent adjunctions plus a projection-formula identification; and the induced maps on the triangulated Grothendieck group. Presented dg unit/counit and cotwist/twist triangles can be compared objectwise, noncanonically, with independently chosen Fourier--Mukai cone triangles; separately supplied comparison data produce natural triangle and functor isomorphisms between the same families. Shift-compatible refinements transfer triangulatedness to the selected Fourier--Mukai twist and cotwist shift structures. The left-adjunction dual twist and dual cotwist expose the same comparison and conditional kernel-autoequivalence APIs through the definitional correspondence swaps. The canonical enhanced twist/cotwist equivalence conditions discharge the ordinary equivalence inputs for the right-adjunction pair without supplying their comparison or shift data. No geometry, and no theorem asserting that a functor is of this form or that it has an adjoint.