Initially dense objects for metrically generated theories

We study initially dense objects for metrically generated constructs X in the sense of [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (5) (2005) 1547–1556 (electronic)]. For the base categories consisting of metrics, quasi-metrics, totally bounded quasi-metrics and totally bounded metrics, a general description of some initially dense objects is given in case the expander is local. Using these results, we will prove that for a large class of metrically generated categories, the base category C can be replaced by the subconstruct consisting of the totally bounded C -metrics.