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.
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.