Lectures on Grothendieck Duality IV: A basic setup for duality

Let there be given, on a category C, a pair (*, *) of adjoint monoidal closed-category-valued pseudofunctors. Thus, to each object X ∈ C is associated a closed category DX , with unit object OX ; and to each C-map ψ : X → Y , adjoint monoidal functors DX ψ∗ ←−− −→ ψ∗ DY . There are also, as before, compatibilities—expressed by commutative diagrams—among adjunction, pseudofunctoriality, and monoidality. The maps giving the monoidal structure on ψ∗ are denoted eψ(E,E) : ψ∗E ⊗ ψ∗E → ψ∗(E ⊗ E′) ( E,E′ ∈ DX ) , νψ : OY → ψ∗OX . Adjoint to the natural composition F ⊗ F ′ → ψ∗ψF ⊗ ψ∗ψF ′ e −→ ψ∗(ψF ⊗ ψ∗F ′) (F, F ′ ∈ DY ) (resp. to νψ : OY → ψ∗OX) we have maps dψ(F, F ′) : ψ∗(F ⊗ F ′)→ ψ∗F ⊗ ψ∗F ′, μψ : ψOY → OX .