Adnan H. Abdulwahid - Department of Mathematics, The University of Iowa (and University of Thi-Qar), ۱۴ MacLean Hall, ۵۲۲۴۲-۱۴۱۹, Iowa City, Iowa, USA.

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.

category, center, comonoid, cocompleteness, co-wellpoweredness