CAUTION: The Sender of this email is not from within Dalhousie. Dear categorists, I have encountered the following pair of thorny questions in the course of an investigation into double categories, to which I would like to know the answers. I would be grateful for any help you may have to offer. Given a horizontal morphism f in a double category A, we may construct a new double category A[f_*] by freely adjoining a vertical companion for f. This new double category comes equipped with a double functor A --> A[f_*], which is the pushout of the inclusion H(2) --> Sq(2) along the double functor H(2) --> A that picks out the horizontal morphism f. (Here H(2) and Sq(2) denote the "free-living horizontal morphism" and the "free-living companion pair" respectively.) Question 1: Is the double functor A --> A[f_*] fully faithful on squares? A related question concerns the "double category of squares" functor Sq : 2-Cat --> DblCat and its left adjoint St : DblCat --> 2-Cat. Question 2: Is the unit double functor A --> Sq(St(A)) fully faithful on squares? Best regards, Alexander [For admin and other information see: http://www.mta.ca/~cat-dist/ ]