Given functions α:A→B, β:B→C, γ:C→D. Then

(1) (γβ)α = γ(βα) (associativity).

(2) If α and β are one-to-one, so is βα.

(3) If α and β are onto, so is βα.

(4) If α is one-to-one and onto, then there is a function α^-1 from B onto A such that (α^-1α)(a) = a for all a in A and (αα^-1)(b) = b for all b in B.