Abstract: Properties can be extended from categories of total maps to partial maps in a uniform way, e.g. cartesian products are lifted to lax cartesian products.
The partial maps of a category A equipped with a dominion M are ordered by their extent of definition, thus forming an ordered category Ptl(A,M). The Ptl functor preserves adjunctions, including those that define products, etc. It has a coreflection Tot that picks out the total maps of an arbitrary ordered category, and a reflection Dom which constructs a category of domains for its morphisms. Each of these adjunctions yields a characterisation of categories of partial maps, without assuming any further structures on the categories.
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