Let $X=\mathbf{A}^1_{\overline{\mathbf{Q}}}-\{0\} = \mathbf{G}_{m,\overline{\mathbf{Q}}}$ be the multiplicative over the field of algebraic numbers. Each finite etale cover $Y\to X$ (with $Y$ connected) is isomorphic to the finite etale morphism $X\to X$ given by $z\mapsto z^n$ for some $n\geq 1$.
The universal covering space $\widetilde{X}$ of $X$ is the projective limit over all finite etale covers of $X$. It's not a scheme. (If $\widetilde{X}$ were a scheme, the "morphism" $\widetilde{X}\to X$ would be an etale morphism with non-finite fibres. That's not possible.)
Is it endowed with a morphism $\widetilde{X}\to X$? In which category should I consider this "morphism"?
Can we describe $\widetilde{X}$ a bit more explicitly using the above description of all finite etale covers of $X$?
One can form the projective limit $\tilde{X}$ in the category of schemes without any problem; it is Spec $\overline{\mathbb Q}(\{z^{1/n}\}_{n \geq 1})$. It is not finite type over $X$, and so in particular is not etale, but so what? There is no theorem saying that a projective limit of finite etale maps is etale.