From a linear context-free grammar simulating an arbitrary instance $(\varphi_1,\varphi_2)$ of the Post correspondence problem PCP we easily construct two nonerasing morphisms $h$ and $g$ with $h$ length-duplicating such that the instance $(h,g)$ of PCP has no solution but $(\varphi_1,\varphi_2)$ possesses a solution if and only if the quotient operation $h\g$ is not injective on its domain. Hence, the undecidability of the injectivity problem follows.

Journal of Automata, Languages and Combinatorics, Volume 6, Number 1, 2001, 91-96