Definition 2

A pointed covering $(E, x)$ of $(B, b)$ is said to be a universal covering if, for every pointed covering $(E', x')$ of $(B, b)$, there exists a unique morphism of coverings of B, $f: E\rightarrow E'$, such that $f(x) =x'$.

If $(E, x)$ and $(E', x')$ are universal coverings of $(B, b)$, the unique B-morphism of $(E, x)$ into $(E', x')$ is an isomorphism of B-spaces.

Let E be a connected covering of B and let $x$ be a point of the fibre $E_b$. Suppose that, for every pointed covering $(E', x')$ of $(B, b)$, there exists a morphism $f: E\rightarrow E'$ of coverings of B such that $f(x) =x'$. Such a morphism $f$ is then unique (I, p. 34, Cor. 1 of Prop. 11), hence $(E, x)$ is a universal covering of $(B, b)$. We shall see later (I, p. 126, Cor. of Prop. 3) that this is in particular the case if every covering of E is trivialisable.