twisted morphism category

I've been reading Lawvere's 1970 paper "Equality in hyperdoctrines and comprehension schema...." which appeared in the AMS series Symposia in Pure Mathematics, volume 17, pp 1-14. In the first line on page 12 he refers to the " 'twisted morphism category' " B^ and the forgetful functor from B^ to (B op) x (B). Can someone explain what this twisted morphism category is and/or provide a reference to the definition? Thanks for your help. Carl Futia