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Corollary 4.6 of this document states the following:

For a topological space $X$ we denote the category of covers of $X$ by $\text{Cov}(X)$. If $X$ and $Y$ are path connected, locally path connected and semi-locally simply connected topological spaces (so that their universal covers exist) such that the categories $\text{Cov}(X)$ and $\text{Cov}(Y)$ are equivalent, then $\pi_1(X)$ is isomorphic to $\pi_1(Y)$.

The proof given has a line which reads "Since $\text{Cov}(X)$ and $\text{Cov}(Y)$ are equivalent, the universal cover of $X$ is equivalent to the universal cover of $Y$."

I am unable to understand what is meant by saying that "the universal cover of $X$ is equivalent to the universal cover of $Y$."

Can somebody please explain this to me. Thanks.

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Since you have an equivalence between the categories $\text{Cov}(X)$ and $\text{Cov}(Y)$ I assume that by "the universal cover of $X$ is equivalent to the universal cover of $Y$" the author means that the universal cover of $X$ (which is an object satisfying a specific universal property) is sent to the universal cover of $Y$ by the said equivalence.

This implies that the automorphisms groups of the two covers are isomorphic, this because an equivalence is a fully-faithful functor, and from that it follows the claim of the corollary: the foundamental groups of the two spaces are isomorphic.

Hope this helps.

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    The universal cover to me seems to be an initial object in the category of *pointed* covers of $(X, x_0)$. But in $\text{Cov}(X)$, what is the universal property that is satisfied by a universal cover? Thanks. – 2017-02-04
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    @caffeinemachine *being an initial object* is a universal property ;) – 2017-02-04
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    Yes of course. But I do not see how a universal cover $p:\tilde X\to X$ is an initial object of $\text{Cov}(X)$. For if $q:Y\to X$ is any cover, then one can find not a unique but a lot of morphisms $\tilde X\to Y$ by sending a chosen point $\tilde x_0\in p^{-1}(x_0)$ to a point $y_0\in q^{-1}(x_0)$. – 2017-02-04
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    @caffeinemachine Sorry I thought objects of $\text{Cov}(X)$ were morphisms between pointed spaces. If you regard covers of $X$ as just continuous maps then it is not hard to see that a universal cover is nothing but a weak initial object in $\text{Cov}(X)$. – 2017-02-04
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    So weak initial objects are preserved under equivalence of categories? – 2017-02-04
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    Yes. The proof is straightforward. – 2017-02-06