  We study cocompleteness, co-wellpoweredness and generators in the centralizer
category of an object or morphism in a monoidal category, and the center or the
weak center of a monoidal category. We explicitly give some answers for when
colimits, cocompleteness, co-wellpoweredness and generators in these monoidal
categories can be inherited from their base monidal categories. Most
importantly, we investigate cofree objects of comonoids in these monoidal
categories.
