Conditions for adjoints

[robin@ucalgary.ca]

(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






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