Definition 9.1 (The category of pointed topological spaces):

This category will be denoted by ${\bf Top_{0}}$ and its objects consists of all pairs $(X, x_0)$ where $X$ is a topological space and $x_0$ is a point of $X$. Given two pairs of pointed spaces $(X, x_0)$ and $(Y, y_0)$, the morphisms between them consists of all continuous functions $f : X \longrightarrow Y$ such that $f(x_0) = y_0$.

Suppose that $X, Y$ are path connected spaces and $f : X \longrightarrow Y$ is a continuous map such that $f(x_0) = y_0$ then $f$ clearly defines a morphism, denoted by same letter, between pointed spaces

$$f:(X, x_0)\longrightarrow (Y, y_0).$$

The map $f_{*}:\pi_1(X. x_0)\longrightarrow \pi_1(Y, y_0)$ given by $f_*([\gamma]) = [f\circ\gamma ]$ where $\gamma$ in $X$ based at $x_0$, is well defined since $f \circ \gamma$ is a loop in $Y$ based at $y_0$ Therefore $f_{\ast}$ is well defined because if $\gamma_1,\gamma_2$ are homotopic loops in $X$ based at $x_0$ and $F$ is the homotopy then $f \circ F$ is a homotopy between $f \circ \gamma_1$ and $f \circ \gamma_2$ in $Y$. It is immediately checked that $f \circ (\gamma_1 \ast \gamma_2)= (f \circ \gamma_1) \ast (f \circ \gamma_2)$ thereby giving a group homomorphism:

$$f_{\ast}([\gamma_1][\gamma_2]) = f_{\ast}([\gamma_1])f_{\ast}([\gamma_2]).$$

The group homomorphism $f_{\ast}$ is called the map induced by $f$ on the fundamental groups. In other words we obtain a functor $\pi_1$ from ${\bf Top}_0$ to ${\bf Gr}$.