Let $X$ be a topological space and $Et/X$ the full subcategory of $Top/X$ of local homeomorphisms. The question is as in the title.
Let $(L \xrightarrow{l} X, \mu)$ be a limiting cone for $M: I \to Et/X$ in $Top/X$. We need to show that $l$ is a local homeomorphism.
If $I$ has no objects, then $l$ is $id_X$ which is obviously a local homeomorphism. Otherwise we can take an object $i$ of $I$. For $x \in L$ the only way to obtain an open neighbourhood of $x$ I can think of is to take a neighbourhood of $\mu_i(x)$ on which $mi: Mi \to X$ is a homeomorphism and then take the preimage under $\mu_i$. However, this fails to be injective if $\mu_i$ is not and we have no reason to assume it is.
I do not see how to use $I$ being finite and $L$ being a limiting cone.
It suffices to show that $Et/X$ has a terminal object and pullbacks by this characterization. It is easy to see that the identity on $X$ gives a terminal object in $Et/X$. So the only thing to verify is that $Et/X$ has pullbacks. By general abstract nonsense, they should be given by a limit in $Top$ of the diagram consisting of the pullback diagram together with the legs to $X$. Hence it suffices to check, that the leg of the limit cone to $X$ is a local homeomorphism (which I did not do yet, so it might fail here but I don’t think so..)