Initial Morphisms versus Embeddings

There is a usual practice in topology of completing an object via an initial morphism XY and then using some ad hoc method of transforming that initial morphism into an embedding XY. In the setting of categorical topology we show that there is a bicoreflective general process available for carrying out such constructions. We further show that this bicoreflector can be adapted to respect a closure operator when the topological construct is endowed with such.