  In this paper, we show that every completely semi-$\phi$-map on a submodule
of a Hilbert $C^*$-module has a completely semi-$\phi$-map extension on the
whole of module. We also investigate the extendability of $\phi$-maps and
provide examples of $\phi$-maps which has no $\phi$-map extension. Finally, we
introduce a category of Hilbert $C^*$-module and determine injective objects in
this category.
