\begin{definition}[Definition:Fundamental Group Functor]
Let $\mathbf{Top}^*$ be the category of pointed topological spaces.
Let $\mathbf{Grp}$ be the category of groups.
The '''fundamental group functor''' is the covariant functor $\pi_1 : \mathbf{Top}^* \to \mathbf{Grp}$ with:
{{DefineFunctor
|ob = A pointed topological space $(X, x_0)$ is sent to its fundamental group.
|mor = A continuous pointed mapping $f$ is sent to its induced homomorphism of fundamental groups $\pi_1(f) = f_*$.
}}
\end{definition}