At the moment we are investigating logical systems and their relations based on the formal concepts of institution (Goguen/Burstall) and entailment system/logic (Meseguer). For our analysis we found appropriate to use concepts like "lax indexed functors" thereby having in mind the corresponding definitions for ordered categories in "Extending properties to categories of partial maps" from Barry Jay [Jay90] (TR ECS-LFCS-90-107). To give an impression about what we are doing, I will give the definition that turns out to be the adequate one for our purposes: A lax indexed functor F from an indexed category C:IND->CAT to an indexed category D:IND->CAT is given by functors F(i):C(i)->D(i) for each i in |IND| and by natural transformations F(g):C(g);F(j)=>F(i);D(g):C(i)->D(j) for each morphism g:i->j in IND such that the following compositionality condition is satisfied for any g:i->j and h:j->k in IND: F(g;h) = (C(g);F(h))*(F(g);D(h)). (We also need the other version where F(g) goes from F(i);D(g) to C(g);F(j).) To get the right feeling and insight we have developed all necessary results by ourselves. Especially we were interested in the generalization of the Grothendieck construction to "lax indexed functors". Now, before fixing these things in a technical report, were are looking for corresponding references of work already done in this direction. Especially we don't want to introduce new names for already known concepts. We need some advice here... Our observation is that we have essentially used for many concepts and results the 2-categorical structure of CAT. Thus we strongly believe that somebody has already defined and investigated "lax functors" and "lax natural transformations" for 2-categories. Thanks for any help. With all best wishes, Alfio Martini.