I'm going to use the language from Lack and Rosicky's Notions of Lawvere theory, but I won't be touching on actual enriched category theory.
Suppose I have a category $\mathbb{C}$ with a class of limits and colimits, $\Phi, \Psi$ respectively, that commute with each other. Now suppose I have a subcategory $\mathbb{C}'$ that is closed to $\Phi$-limits, are there any conditions under which the $\Psi$-cocompletion of $\mathbb{C}'$ in $\mathbb{C}$ will be $\Phi$-complete?
