Definition 19.3:

Let us consider a fixed connected topological space $X$ with a specified base point $x_0 \in X$. A homomorphism between two coverings $p:(Y, y_0)\longrightarrow (X, x_0)$ and $q:(Z,z_0)\longrightarrow (X, x))$ is a surjective continuous map $r:(Y, y_0)\longrightarrow (Z, z_0)$ such that $q\circ r = p$ or diagrammatically,

$$\xymatrix{ (Y, y_0) \ar[rr]^{r}\ar[rd]_{p} & & (Z, z_0) \ar[ld]^{q}\\ & (X, x_0) }$$

The definition enables us to form a category of coverings of a given space $X$ with a specified base point $x_0 \in X$. To obtain a satisfactory theory one must impose some additional assumption on $X$ such as local connectedness. In other words $r$ is a lift of $p$ with respect to the covering map $q$. The universal covering is then defined in terms of a universal property.