Dear Jean On 15 Feb 2017, at 4:47 PM, Jean Benabou <jean.benabou@wanadoo.fr<mailto:jean.benabou@wanadoo.fr>> wrote: However they both say that if S is a category with finite limits the 2-category Cat(S) is what I called strictly finitely 2-complete. I doubt it. Could they, or anybody else, tell me how yo prove Cat(S) has strict inverters? The type of limit of which you speak for 2-categories are those of ordinary enriched category theory with base Cat. First agree that Cat(S) has terminal object, pullbacks and cotensoring with the arrow category Arr (usually called blackboard bold 2). Next construct the category Iso with two objects and an isomorphism between them using finite colimits in Cat: take the pushout of the two functors 2 --> Arr which are bijective on objects to obtain the category with two objects and an arrow each way; then force the arrows to be inverse to each other using two coequalizers. This tells us how to obtain the cotensor {Iso, A} of Iso with any A in Cat(S) by a pullback and two equalizers. Now take a 2-cell t between two morphisms f, g : A --> B in Cat(S). It corresponds to an morphism t' : A --> {Arr, B} in Cat(S). The inverter of the 2-cell t is the pullback of the restriction morphism {Iso, B} --> {Arr, B} along t' : A --> {Arr, B}. These things go back to my paper 10. Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8 (1976) 149--181; MR53#5695. (Now I prefer the term weighted limit to indexed limit.) For any nice base for enrichment, all weighted limits can be obtained from products, equalizers and cotensoring. In the case of Cat, cotensoring with Arr suffices. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]