Let $V$ be a symmetric monoidal category which is complete and cocomplete. Is the category of small symmetric colored $V$-enriched operads complete and cocomplete? If $V$ is presentable, is it presentable?
In the case of $V$-enriched categories, corresponding results are proven in the following papers:
- Wolff, $V$-cat and $V$-graph.
- Kelly, Lack, $V$-Cat is locally presentable or locally bounded if $V$ is so.