Let T: X → Y be a transformation. Recall that T is onto (surjective) if its range is the whole target set. More specifically, this means
For any y ∈ Y, there is x ∈ X, such that T(x) = y.
The following characterizes onto transformations. More properties can be found in this exercise.
Proof If TS = idY, then for any y ∈ Y, the element x = S(y) satisfies T(x) = TS(y) = idY(y) = y. This proves that any element of Y is in the range.
Conversely, suppose T is onto. Then for any y ∈ Y, there is x ∈ X, such that (x, y) ∈ T. We choose S(y) to be one of such x. After S(y) is chosen for all y ∈ Y, we have a transformation S: Y → X. By the way S is constructed, we clearly have TS = idY.