It is proved that if $\nu_1$ and $\nu_2$ are two computable numerations of a certain family of recursively enumerable sets such that $\nu_2<_p\nu_1$ and $\nu_1$ is not a $p$-principal numeration, then there exists a computable numeration $\nu_0$ p-incomparable with $\nu_1$ such that $\nu_2<_p\nu_0$. This yields the description of injective objects and the absence of numerated sets projective in the category $K_p$ conforming to $p$-reducibility of computable numeration.