The doubly infinite diagram of exact sequences that an additive bifunctor T associates with a pair of short exact sequences can be regarded as a web diagram lying on the surface of a cylinder. The diagram has six diagonal sequences involving two graded derived functors that arise through the failure of T to preserve pull-backs respectively push-outs. In the case T = Hom(-,-) one of the diagonal sequences is equivalent to the bivariant Hom-Ext sequence studied by Pressmann [4].