Diagram chasing in ordered categories with involution

To carry out diagram chases in non-additive categories, we make use of cate gories of relations, ordered categories with involution. For illustration, we give two constructions of the connecting homomorphism. In the top-down t reatment, morphisms are just relations with right adjoints. In the bottom-u p treatment for a category with a given class of regular epimorphisms, we c onstruct relations following a method of Calenko. (C) 1999 Elsevier Science B.V. All rights reserved. AMS Classification: 18B10; 18G99.