Definition:Fundamental Group Functor

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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:

Object functor:         A pointed topological space $(X, x_0)$ is sent to its fundamental group.
Morphism functor: A continuous pointed mapping $f$ is sent to its induced homomorphism of fundamental groups $\pi_1(f) = f_*$.

Also see