limits and colimits of enriched categories

In Kelly's book on enriched category theory, he seems to claim that it would be easy to prove that if K is a symmetric monoidal closed category in which all small diagrams have limits and colimits, then in the category of small K-categories it is also true that all small diagrams have limits and colimits, but he does not actually prove this. Does anyone know a reference for a proof of this fact? John Baez