In detail, a pointed set is a pair $\webleft (X,x_{0}\webright )$ consisting of:
- The Underlying Set. A set $X$, called the underlying set of $\webleft (X,x_{0}\webright )$.
- The Basepoint. A morphism
\[ \webleft [x_{0}\webright ]\colon \text{pt}\to X \]
in $\mathsf{Sets}$, determining an element $x_{0}\in X$, called the basepoint of $X$.
