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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.

An isomorphism of categories is a functor with a strictly inverse functor. It is equivalently bijective on objects and on each hom-set. An equivalence of categories only requires inverse composites up to invertible natural transformations. For example, the category of partial functions and the category of pointed sets are equivalent but their actual object collections prevent an isomorphism: the former has one zero object, the empty set, while the latter has distinct singleton pointed set objects that are all zero objects.

A pointed set is a set $X$ with a distinguished element $x_0\in X$. A morphism of pointed sets is a function carrying the distinguished element to the distinguished element. The resulting category of pointed sets has every singleton as a zero object.