I have a question about some hypothesis in a classical theorem about coverings.
The theorem is the following:
Let $B$ be a connected base space, and $X$ and $Y$ be coverings of $B$ and $g, f: X\to Y$ two $B$-morphisms. If there exists $b\in B$ such that $f_{|X_b} = g_{|X_b}$ then $f=g$.
This is easy to prove is $X$ is connected. If you don't assume $X$ connected, then it is also easy to prove if you assume $B$ locally connected, as in that case every connected component of $X$ is open and closed, and therefore maps surjectively onto $B$ and thus, every connected component of $X$ contains a point of the fiber $X_b$.
This is of course a standard assumption.
But the reference I'm looking at (Douady, Algèbres et théorie galoisiennes) does not make any assumption (other than $B$ connected) and simply states that "every connected component of $X$ contains a point in the fiber $X_b$".
And I fail to see how to prove that whithout the local connectedness assumption, am I missing something?