exercise: what is the right adjoint of the functor L : C^2 -> C^3 which sends (x, y) to (a*x + b*y, c*x + d*y, e*x + f*y) ?

answer: R : C^3 -> C^2 sends (u, v, w) to (u^a * v^c * w^e, u^b * v^d * w^f).

the unit 1 -> R . L has components
  (x, y) -> ((a*x + b*y)^a * (c*x + d*y)^c * (e*x + f*y)^e,
             (a*x + b*y)^b * (c*x + d*y)^d * (e*x + f*y)^f)
given in the obvious way,

while the counit L . R -> 1 has components
  (a*(u^a * v^c * w^e) + b*(u^b * v^d * w^f),
   c*(u^a * v^c * w^e) + d*(u^b * v^d * w^f),
   e*(u^a * v^c * w^e) + f*(u^b * v^d * w^f)) -> (u, v, w)
likewise.