Dear categorists, I am wondering if the following property of a functor U : C -> D has a name in the literature: * For every lifting problem in C and any solution in D to the image under U, there is a unique solution in C whose image under U is that solution. More precisely: * For any morphisms X -> Y and Z -> W in C, the induced commutative diagram C(W, X) ------> C(Z, X) \times_{C(Z, Y)} C(W, Y) | | | | v v D(UW, UX) --> D(UZ, UX) \times_{D(UZ, UY)} D(UW, UY) is a pullback square. Of course, any fully faithful functor has the property in question; a less trivial example is the projection from a (co)slice category to its base. Every functor between groupoids has this property, so they need not be faithful. One also notes that the class of functors with this property is closed under composition. It is not hard to see that if a functor has the above property, then it reflects both orthogonality and weak orthogonality in the naive sense. The converse is false. Nonetheless, my inclination is to call these functors "orthogonality-reflecting". Best wishes, -- Zhen Lin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]