Definition 1.5.18.  Let X, Y be sets, f : X → Y be a function. We define:
f is bijective  :⇔  f is injective and f is surjective  ⇔  ∀ y ∈ Y : ∃! x ∈ X : f(x) = y  ⇔  ∃ g : Y → X : [g ∘ f = idX and f ∘ g = idY]  ⇔  ∃! g : Y → X : [g ∘ f = idX and f ∘ g = idY]
Equivalence.  No proof.
References.