This paper investigates the completeness and cocompleteness of some categories of Yoneda complete metric spaces. It is shown that if the morphisms are chosen to be Yoneda continuous maps or Yoneda continuous nonexpansive maps, then the category is both complete and cocomplete; if the morphisms are chosen to be Yoneda continuous Lipschitz maps, then the category is finitely complete and finitely cocomplete, but neither complete nor cocomplete. It is also shown the category of real-valued continuous lattice and Yoneda continuous right adjoints is complete.
