In answer to Hendryk Pfeiffer Internal categories have been extensively studied in the late sixties and seventies around Charles Ehresmann, both by themselves and as a particular case of 'sketched structures'. Many of the papers of Charles are devoted to this subject, and they are re-printed in "Charles Ehresmann: Oeuvres.completes et commentees" Parts III and IV. In particular, in our joint paper "Categories of sketched structures" (Cahiers de Top. et Geom. Diff. XIII-2, 1972, reprinted in the "Oeuvres" Part IV-2, 407-516) there is a general study and, in Part III * C is symmetric monoidal and closed
Maybe the following point of view (which views Cat(C) as a 1-category) is more familiar. If one studies essentially algebraic theories, one writes,
(1) Cat(C)=3DLEFunc(Th(Cat),C).
Here LEFunc(D,C) is the category of finite-limit preserving functors D->C and natural transformations. C,D are categories with finite limits. Th(Cat) is a suitable category with finite limits (`theory of categories') such tha= t (1) holds. Cat(C) is then called a model of the theory of categories in the category C.
Then my above questions come down to the following: If C is enriched/monoidal/..., what does this imply for LEFunc(D,C)? I am also interested in the 2-categorical structure of Cat(C) which is not visible in this picture.
I guess all this is known and has been written somewhere. But where? I woul= d appreciate any sort of comments.
Hendryk Pfeiffer