Equivalent characterizations of Henselian Rings (Theorem 4.2 in James Milne's "Étale Cohomology")

I am stuck on a step in the proof of Theorem 4.2 in Chapter I of James Milne's "Étale Cohomology". The particular implication is (c) $\Rightarrow$ (d).

Let $X=\text{Spec} (A)$, where $A$ is a local ring with the maximal ideal $m$, and let $x$ be the unique closed point of $X$.

(c) If $f:Y\to X$ is a quasi-finite separated morphism, then $Y=Y_0\coprod Y_1\coprod\ldots\coprod Y_n$, where $f(Y_0)$ does not contain $x$ and $Y_i$ are spectrums of local rings and are finite over $X$ for $i\geqslant 1.$

(d) For any étale morphism $g:Y\to X$ such that there is a point $y\in Y$ such that $f(y)=x,\ k(y)=k(x)$ there is a section $s: X\to Y.$

The proof states:

Using (c), we may reduce the question to the case of a finite étale local homomorphism $A\to B$ such that $k(m)=k(n)$, where $n$ is the maximal ideal of $B$. According to (2.9b), $B$ is a free $A$-module, and since $k(n)=B\otimes_A k(m)=k(m)$ it must have rank 1, that is, $A\approx B.$

I think everything makes sense if we add the separated hypothesis to (d). As I understand the proof in this case, we are applying (c), restricting to the $Y_i$, and then we find at least one of them such that there is a $y\in Y_i$ with $f(y)=x,\ k(y)=k(x)$, and finally use flatness to argue that some $Y_i$ is isomorphic to $X$. But without this additional hypothesis, I don't see how (c) can be applied.

Yes, the morphism in (d) should be assumed separated. This appears to be an oversight on the author's part.

Sorry I bothered trying to help. I'll refrain in future.

Note, this answer was deleted almost instantly by the vigilantes

José Carlos Santos, Harish Chandra Rajpoot, amWhy,

but the the OP found it helpful, so I put it back up.