Category of Pointed Sets is Category

We show that composition of two pointed maps yields a pointed map, and that the identity is a pointed map.

Let $(A, a) \xrightarrow{f} (B, b) $ and $(A, a) \xrightarrow{g} (B, b) $ be arrows of $\mathbf{Set}_*$.

By the definition of morphisms in $\mathbf{Set}_*$We see $(g \circ f) (a) = g(b) = c$, whence $g \circ f$ is a pointed morphism from $(A,a)$ to $(C, c)$.

The identity map on $A$ is evidently a pointed map for any choice $a \in A$ of base point, and so serves as identity for any object $(A,a)$.

That pointed mappings are associative follow from the fact that their composition is just composition of set maps.