Corollary 2

Let $(E, p)$ and $(E', p')$ be connected coverings of B. Let $x$ be a point of $E_b$ and $x'$ a point of $E'_b$. In order that there exist a B-morphism $g: E\rightarrow E'$ such that $g(x) =x'$, it is necessary and sufficient that one have $p_*(\pi_1(E, x))\subset p'_*(\pi_1(E', x'))$. Such a morphism is then unique, and is an isomorphism if and only if the subgroups $p_*(\pi_1(E, x))$ and $p'_*(\pi_1(E', x'))$ of $\pi_1(B, b)$ are equal.

According to III, p. 305, Theorem 1, the mapping of $\pi_1(B, b)$ onto $E_b$ defined by $\gamma \mapsto x\cdot \gamma$ induces by passing to the quotient an isomorphism of the $\pi_1(B, b)$-set $p_*(\pi_1(E, x))\backslash \pi_1(B, b)$ onto $E_b$. Analogously, there exists a unique isomorphism of $\pi_1(B, b)$-sets of $p_*(\pi_1(E', x'))\backslash \pi_1(B, b)$ onto $E'_b$. According to Proposition 2, there exists a B-morphism $g: E\rightarrow E'$ such that $g(x) =x'$ if and only if there exists a morphism of $\pi_1(B, b)$-sets of $p_*(\pi_1(E, x))\backslash \pi_1(B, b)$ onto $p'_*(\pi_1(E', x'))\backslash \pi_1(B, b)$ which sends the class $p_*(\pi_1(E, x))$ onto the class $p'_*(\pi_1(E', x'))$. Such a morphism exists if and only if one has $p_*(\pi_1(E, x))\subset p'_*(\pi_1(E', x'))$. It is then unique, since the space E is assumed connected (I, p. 34, cor. 2 of prop. 11). It is an isomorphism if and only if the subgroups $p_*(\pi_1(E, x))$ and $p'_*(\pi_1(E', x'))$ of $\pi_1(B, b)$ are equal.