  It is argued in (Eklund et al., 2018) that the quantale [L,L] of
sup-preserving endomaps of a complete lattice L is a Girard quantale exactly
when L is completely distributive. We have argued in (Santocanale, 2020) that
this Girard quantale structure arises from the dual quantale of inf-preserving
endomaps of L via Raney's transforms and extends to a Girard quantaloid
structure on the full subcategory of SLatt (the category of complete lattices
and sup-preserving maps) whose objects are the completely distributive
lattices.
  It is the goal of this talk to illustrate further this connection between the
quantale structure, Raney's transforms, and complete distributivity. Raney's
transforms are indeed mix maps in the isomix category SLatt and most of the
theory can be developed relying on naturality of these maps. We complete then
the remarks on cyclic elements of [L,L] developed in (Santocanale, 2020) by
investigating its dualizing elements. We argue that if [L,L] has the structure
a Frobenius quantale, that is, if it has a dualizing element, not necessarily a
cyclic one, then L is once more completely distributive. It follows then from a
general statement on involutive residuated lattices that there is a bijection
between dualizing elements of [L,L] and automorphisms of L. Finally, we also
argue that if L is finite and [L,L] is autodual, then L is distributive.
