Heights and morphisms in number fields

We give a formula with explicit error term for the number of $K$-rational points $P$ satisfying $H(f(P))\le X$ as $X\to \mathrm{\infty }$, where $f$ is a nonconstant morphism between projective spaces defined over a number field $K$ and $H$ is the absolute multiplicative Weil height. This yields formulae for the counting functions of $f({\mathbb{P}}^m(K))$ with respect to the Weil height as well as of ${\mathbb{P}}^m(K)$ with respect to the Call-Silverman canonical height.