= Solution
A <conformal map> preserves oriented <angles> between regular smooth curves through each point. For a <holomorphic> map with nonzero <derivative>, the <derivative> also describes the local scale and <rotation>.
Indeed, if two curves $\gamma_1,\gamma_2$ meet at $z$ and have nonzero tangent <vectors> $v_1,v_2$, the real <chain rule> and <complex differentiability> give
$$
(f\circ\gamma_j)'=f'(z)v_j.
$$
Because $f'(z)\neq0$, the <image> <tangents> are nonzero and
$$
\boxed{\frac{(f\circ\gamma_2)'}{(f\circ\gamma_1)'}=\frac{v_2}{v_1}.}
$$
Taking arguments proves preservation of oriented <angles>. Multiplication by $f'(z)$ rotates every tangent by $\arg f'(z)$ and scales every length by $|f'(z)|$. Thus the hypothesis gives <conformality> at every point of $U$; global <injectivity> is not needed for this local conclusion.
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