Dear Philippe On 14 Apr 2017, at 12:13 AM, gaucher <gaucher@irif.fr<mailto:gaucher@irif.fr>> wrote: 1) Let K be a complete, cocomplete and cartesian closed category. Consider the category DK of all small diagrams over K. The objects are all small diagrams F:I-->K from a small category I to K. And a map from (F:I-->K) to (G:J-->K) is a functor f:I-->J together with a natural transformation mu:F-->Gf. DK is complete and cocomplete and I would like to know if it is cartesian closed as well. Yes it is. The internal hom of F : I --> K and G : J --> K is H : [I,J] --> K defined by Hr = end over i in I of [Fi,Gri] where [I,J] is internal hom in Cat and, for h, k in K, [h,k] is internal hom in K. 3) I also would like to know what is known about the link between locally presentability and fibred category. I suspect that, if T : K^{op} --> LFP is a functor, where K is locally finitely presentable (lfp) and LFP is the 2-category of lfp categories and right adjoint functors which preserve filtered colimits, then the ``Grothendieck fibration construction'' El(T) --> T gives an lfp category El(T). I can't think of a quick reason but others may think of one, a reference, or a counterexample. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]