Interpreting category-valued presheaves

Dear categorists, I could not find an answer to the following, so I need your help: Given a category C we know that the category of set-valued presheaves on C may be interpreted as a free cocompletion of C. More precisely, if we denote PSh(C) the presheaf category, then colimit-preserving functors from PSh(C) to a cocomplete category D correspond to functors from C to D (via Yoneda extension). I am interested in an interpretation of category-valued presheaves. Is there some sort of 2-categorical version of the above fact? Best regards, Redi Haderi [For admin and other information see: http://www.mta.ca/~cat-dist/ ]