First, may I ask that we please not get into name-calling or ad hominem attacks. You should think of this as a site at a professional level, and the behavior should be more or less that expected at a professional seminar.
The question is, I think, quite reasonable. A limit or colimit as seen from within a countable model of set theory may not be the limit or colimit as seen from an external point of view. But note that the "external" judgment itself involves a background model $V$!
None of this should be a worry. The point is that, for any reasonable theory of sets, limits and colimits exist and are unique up to isomorphism for any small diagram which is definable in the theory. Each model of the theory will interpret the terms as they will, and each model is complete and cocomplete according to that interpretation.
Other categories such as $Grp$, and forgetful functors and so on, are definable by class formulas we can write down in the theory, and these too will be interpreted as they will for each individual model. We understand that the "meanings" of these terms are model-dependent, but in any event it's enough to recognize that the theory itself is expressive enough to accommodate limits and colimits, etc.