ct.category theory – boundaries, colimits and universes

In category theory, limits and colimits of charts are considered for many purposes $$F colon mathsf {J to C}$$ from where $$mathsf {J}$$ is a small category, ie a category in which the classes of objects and morphisms are actually sets.

But what if we want to adopt the foundation system with Grothendieck universes instead of sets and classes? Do we need the trifling condition at all?

For example, in classes (without universes) we say a category $$mathsf {C}$$ is complete (or cocomplete) if there is a limit (or colimits) of all charts indexed by small categories. What are the appropriate versions of these terms in universes?