Abstract
A general kind of morphism of diagrams in a category is introduced. It is the most general notion of morphism which induces a morphism between the colimits of the diagrams. The sense in which it is the most general is made precise. It is expressed in terms of total profunctors which generalize everywhere defined relations. Their functorial properties are developed leading to the notion of cohesive family of diagrams. A complementary notion of deterministic profunctor is also introduced generalizing single valuedness.
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