Category of partial functions ($\mathbf{Part}$)

The category $\mathbf{Part}$ has sets as objects and partial functions as morphisms. Composition is defined where both successive functions are defined. The nowhere-defined map is a zero morphism, and the empty set is its sole actual zero object. Adjoining a tagged basepoint turns a partial function into a total basepoint-preserving function, giving an equivalence of categories with the category of pointed sets.