Category of sets

Example

The category of sets, written $\mathbf{Set}$, has these parts. Its objects are all sets. A morphism from $A$ to $B$ is a function $f \colon A \to B$. Composition is the composition of functions, $(g \circ f)(a) = g(f(a))$, and the identity of $A$ is the identity function $\id_A$.

This is a category. Composition of functions is associative, since both $(h \circ g) \circ f$ and $h \circ (g \circ f)$ send $a$ to $h(g(f(a)))$. The identity laws hold, since $f \circ \id_A$ and $\id_B \circ f$ both send $a$ to $f(a)$. The collection of all sets is not a set, so $\mathbf{Set}$ is a large category. Each hom-set $\mathbf{Set}(A, B)$ is a set, so $\mathbf{Set}$ is locally small.