Dear category theorists, I have a question in the context of 'biclosed' monoidal 2-categories. To give an analogy, let us begin with one dimension less, i.e., with a closed monoidal category C. Given an arbitrary object X \in C it is then easy to show that the internal hom-object END(X)=HOM(X,X) is canonically a monoid and, in fact, the terminal example of a monoid acting on $X$. One way to make the latter result precise is as follows. The category Mod(C) of modules in C, where objects are pairs (M,Y) consisting of a monoid M and a left M-module Y is endowed with a forgetful functor U:Mod(C)-->C. This functor U can also be obtained by first observing that the monoidal unit S can be endowed with the structure of a monoid making it the initial monoid and such that the category S-Mod is isomorphic to C. U is then induced by restriction of scalars along the unique monoid morphisms S-->M and the isomorphism S-Mod\cong C. The statement that the canonical action of END(X) on X is the terminal example of a monoid acting on X is now precisely the statement that it gives us a terminal object of U^{-1}(X). I would now love to have corresponding results in the following 2-categorical situation where there are two 'degenerations': the closedness of the monoidal structure is not expressed by a 2-adjunction but only by a 'biadjunction' and we consider pseudo-monoids instead of monoids. Thus, let us consider a symmetric monoidal 2-category C which is 'biclosed' in the sense that for every X there is a right biadjoint 2-functor to -\otimes X, i.e., we have an internal hom 2-functor HOM(X,-) and natural equivalences of categories Hom(W\otimes X,Y)-->Hom(W,HOM(X,Y)) (where Hom is the enriched hom of C). I would now love to have the following results: i) For an arbitrary object X \in C, the internal hom END(X)=HOM(X,X) can be canonically endowed with the structure of a pseudo-monoid. ii) There is a canonical action of the pseudo-monoid END(X) on X induced by the 'biadjunction counit'. iii) This action is 'the' 'biterminal' example of such an action: using a 2-categorical version of the Grothendieck construction one can form the 2-category PsMod(C) of pseudo-modules in C which is again endowed with a projection functor U:PsMod(C)-->C. Given an object X \in C the canonical action of ii) is then a bi-terminal object of U^{-1}(X) in the sense that all hom-categories of morphisms into that object are equivalent to the terminal category. It would be of great help if someone could give me a reference to the literature where such issues are discussed. Just in case that someone of you has too much time I would also like to get something close to iv) The monoidal S unit is 'the' initial example a of pseudo-monoid and we have an equivalence of categories S-PsMod \simeq C but i)-iii) are more important to me. Thanks a lot in advance! Best, Moritz Groth. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]