Category of sets ($\mathbf{Set}$)

The category of sets has sets as objects and functions as morphisms.

The category whose objects are finite sets and whose morphisms are all functions between them. Finite categorical limits and colimits are the ordinary set constructions. It is essentially small: a skeleton of a category consists of the sets $\{0,\dots ,n-1\}$ for $n\in \mathbb{N}$. Its Yoneda embedding preserves limits but generally does not preserve coproducts: evaluating $y1⨿y1$ at a two-element set gives two elements, while evaluating $y2$ gives four.

The category of pointed sets and basepoint-preserving functions is a pointed category. It is equivalent to the category of partial functions by adjoining or deleting the basepoint. It is not isomorphic to the category of all actual sets and partial functions: there are many distinct singleton zero objects, whereas the empty set is the sole zero object in that category.

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 endofunctor $F:\mathbf{Set}\to \mathbf{Set}$ preserving finite limits and finite colimits has a unique natural monomorphism $\alpha :1_{\mathbf{Set}}\to F$. It also preserves countable coproducts. If some ${\alpha }_A$ is not surjective, the extra element determines a countably complete nonprincipal ultrafilter on $A$.

For an ultrafilter $U$ on $A$, the assignmentis an endofunctor of sets. It preserves finite limits. If $U$ is countably complete, every map $A\to {\coprod }_n{B}_n$ lands in one summand on a $U$-large set, so $F$ also preserves countable coproducts.