what is unique inverse?

f: X to Y is invertible.

this means, f has an inverse

let g: Y to X be inverse of f = f  ^-1

g o f = f o g = I

let inverse is not unique. let   g₁ : Y to X be another inverse of f

g₁ o f =f o g₁ = I

g₁ o f =g o f

composing both sides with g, we get

(g₁ o f) o g = (g o f) o g

g₁ o (f o g) = g o (f o g)

g₁ o I = g o I

g₁ = g

hence, inverse is unique.