Thanks to Toby Bartels I have got now an idea how to translate "prone arrow" and "supine arrow" to German, namely as "Pfeil in Bauchlage" and "Pfeil in Rueckenlage" (in Engl. "arrow lying on the belly" and "arrow lying on the back"). At least in German such terminology would be "frowned upon" and actually it sounds very strange. I haven't got a feeling how strange it sounds in English (but I guess it does!). The "linguistic" problem I have with this terminology is that I don't know what is the "belly" or "back" of an arrow. The intention of the suggested terminology (besides sounding funny) seems to be to replace vertical/cartesian by vertical/horizontal. But then one is sort of forced to call cocartesian arrows "cohorizontal" which sounds a bit like "vertical", isn't it? But actually, the "co" here refers to the fact that there are 2 ways of being horizontal to all vertical arrows (namely a left and a right one because the orthogonality relation is not symmetric). So one might be inclined to call "cartesian" "right horizontal" and cocartesian "left horizontal". For prefibrations and precofibrations one knows that right horizontal coincides with cartesian and left horizontal with cocartesian, respectively. But, unfortunately, for defining pre(co)fibrations one rather needs the notion of pre(co)cartesian arrow which cannot be characterized in terms of orthogonality conditions. As pointed out by Jean for arbitrary functors (already prefoliating ones) horizontal arrows need not be even precartesian (e.g. when all fibres are discrete). Finally, if one prefers to call pullbacks cartesian squares then one might be inclined to prefer the terminology "cartesian" because the cartesian arrows of the fundamental (also often called codomain) fibration are just the pullbacks, i.e. the cartesian squares. I rather have a different problem with "traditional" terminology, namely that sometimes what Jean in his mail called cartesian and precartesian is called hypercartesian and cartesian Usually, i.e. when studying fibrations, this is not a problem because all notions coincide. But if one considers prefibrations the terminology cartesian/precartesian has the advantage that one speaks about existence of cartesian liftings when defining fibrations and about precartesian liftings when speaking about prefibrations. Thomas Streicher