Hello, Given a category C, I am interesting in the category DC defined as follows: an object is a functor F:I-->C where I is a small category, and a morphism from F:I-->C to G:J-->C is a functor from phi:I-->J together with a natural transformation from F to G o phi. If C is locally small (resp. complete, cocomplete, cartesian closed), then so is DC (see CTGDC vol XVIII-4 (1977) "Decompositions et lax-completions", Guitart et Van Den Bril). My question (for today) is : consider the "forgetful functor" from DC to Cat (the category of small categories) sending F:I-->C to I. Does this functor commute with colimits ? The paper above seems to claim that the commutativity holds if the colimit is a coproduct (see p376). What about general colimits ? Thanks in advance for any help or any other pointer for DC. pg. 25-Nov-2002 20:24:28 -0400,1614;000000000000-00000000