Is not the result you seek (4.56) of Kelly's Basic Concepts of Enriched Category Theory, in the case where $\mathcal V = \mathbf Cat$ ? On Sun, Feb 16, 2020, 11:33 Redi Haderi <haderiredi@gmail.com> wrote:
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
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