Theorem 18.2 (Uniqueness of simply connected covers):

Suppose that $p_1:({\tilde X}_1, {\tilde x}_1)\longrightarrow (X, x_0)$ and $p_2:({\tilde X}_2, {\tilde x}_2)\longrightarrow (X, x_0)$ are covering projections such that both ${\tilde X}_1$ and ${\tilde X}_2$ are simply connected and locally path connected. Then there is a homeomorphism $\psi:{\tilde X}_1\longrightarrow {\tilde X}_2$ such that

$$p_2\circ \psi = p_1. \quad {INSERT DIAGRAM}$$