Construction of a conformal map

Question

Construct a conformal mapping which maps the unit disc centered at the origin bijectively onto the whole complex plane minus the positive imaginary axis $\{iy : y \geq 0\}$ and which additionally satisfies $$\lim_{z\to 1} f(z) = 0,\ \text{and}\ f(0) = i.$$ My effort: Because of the given conditions, I considered $$f(z)=\frac{z-1}{z+i}.$$ I have check this above map is conformal, one-one and satisfying the required conditions. But I am doubtful about bijectivity of this map. Have I started correctly? If not, please correct me.