(Apologies to those who received the earlier type mashed version ...) Jeremy Dawson and I were discusing whether one can express the conditions for an adjoint without requiring functors ... this is what we came up with: There is an adjoint between two categories if and only if there are object functions F and G (not functors) and for each X in \X and Y in \Y there are functions: #: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat between the homsets such that (1) @(#(1)) = 1 and dually #(@(1)) = 1 (inverse on identities) (2) @(1) @(#(1) #(f)) = f and dually #(@(g) @(1)) #(1) = g (3) @(#(f @(1)) h k) = f @(h) @(#(1) k) and dually #(x y @(#(1) z)) = #(x @(1)) #(y) z. I find it hard to believe that such conditions have not been recorded. Does anyone have a reference or similar conditions which do not require functors? -robin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]