 Category of finite sets

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.