A poset $P$ is complete and cocomplete as a category if and only if it is a complete lattice, that is, if and only if every subset $A \subset P$ has both an infimum $($greatest lower bound$)$ and a supremum $($least upper bound$)$.
In any poset (or preorder), a limit of a diagram is an infimum of its objects, while a colimit of a diagram is a supremum of its objects. Whether or not there are any morphisms in the diagram makes no difference because all diagrams in a preorder commute. In particular, any collection of morphisms with common domain defines a cone over any diagram whose objects are indexed by the codomains.