Presheaves etc. in a uniform way Set^C is normally produced directly from the small category C and the category Set as a functor category. Likewise Chu(Set,\Sigma) is produced from the set \Sigma and the category Set via the Chu construction. We give here a uniform way of producing all presheaf categories at once, or all Chu(Set,-) categories at once, without taking any of Set, C, or \Sigma as explicit parameters. This generalizes my note "Chu(Set,K) without sets" of a week ago, in the process answering and extending question (iii) of http://boole.stanford.edu/pub/couple.pdf Let G,K be two sets (of object symbols). Define a GK-category (C,g,k) to consist of a locally small category C and a pair of maps g:G->ob(C), k:K->ob(C) (interpreting object symbols as objects of C). Define a GK-functor F:(C,g,k)->(C',g',k') to be a functor F:C->C' between GK-categories satisfying Fg=g', Fk=k' and fully faithful on homsets from im(g) and homsets to im(k). Write GK-CAT for the 2-category of GK-categories and GK-functors. (GK-Cat is the same with every C small rather than merely locally small.) Associate to each (C,g,k) the full subcategories \G, \K of C having as objects those of im(g), im(k) respectively, together with a GxK matrix Q of cardinals giving the cardinality |C(g(i),k(j))| for i,j in G,K. These associated entities are all preserved up to isomorphism by GK-functors. GK-CAT thus partitions as a sum of connected components over all isomorphism classes of categories \G and \K with respectively |G| and |K| objects, and all GxK matrices of cardinals. It is a straightforward exercise to show that each such connected component of GK-CAT has a 2-final object, which we may call the locally 2-final objects of GK-CAT. These objects are equivalent to the following under the stated conditions. Set |G| = 1, |K| = 0, \G = 1 (the one-morphism category) Set\op |G| = 0, |K| = 1, \K = 1 Set x Set\op |G| = 1, |K| = 1, \G = 1, \K = 1, Q[0,0] = 1 Set^{M\op} (M-sets) |G| = 1, |K| = 0, M = \G (as a one-object category) (Set^M)\op |G| = 0, |K| = 1, M = \K Set^{C\op} (presheaves) |K| = 0, C = \G All presheaf categories |K| = 0 (Set^D)\op (dual preshvs) |G| = 0, D = \K Set^{C\op}x(Set^D)\op C = \G, D = \K, Q[i,j] = 1, i,j in G,K Chu(Set,\Sigma) |G| = |K| = 1, \G = \K = 1, Q[0,0] = \Sigma All Chu(Set,-) categories |G| = |K| = 1, \G = \K = 1 "Presheaf Chu" No restrictions The difference between Set^{C\op} x (Set^D)\op and Presheaf Chu is that the restriction Q[i,j] = 1 (meaning that homsets from im(g) to im(k) have exactly one morphism) of the former nullifies the effect of the Chu matrices (one matrix for the ordinary Chu construction). When this restriction is dropped the notion of continuity enters, with ordinary topological continuity at Q[0,0] = 2 and more general Chu continuity obtaining for larger cardinals in Q. It is plausible that a necessary and sufficient condition for a presheaf-Chu category to be *-autonomous is for \G and \K to be isomorphic, as a generalization of \G = \K = 1. Vaughan Pratt