  Let $k$ be a field with separable closure $\bar{k}\supset k$, and let $X$ be
a qcqs $k$-scheme. We use the theory of profinite Galois categories developed
by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences
\begin{equation*} \Pi_{<\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to
\Pi_{<\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \qquad
\text{and} \qquad \widehat{\Pi}{}_{\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}})
\to \widehat{\Pi}{}_{\infty}^{\mathrm{\acute{e}t}}(X) \to
\mathrm{BGal}(\bar{k}/k) \end{equation*} of protruncated and profinite \'etale
homotopy types are fiber sequences. This gives a common conceptual reason for
the following two phenomena: first, the higher \'etale homotopy groups of $X$
and the geometric fiber $X_{\bar{k}}$ are isomorphic, and second, if
$X_{\bar{k}}$ is connected, then the sequence of profinite \'etale fundamental
groups
$1\to\hat{\pi}{}_{1}^{\mathrm{\acute{e}t}}(X_{\bar{k}})\to\hat{\pi}{}_{1}^{\mathrm{\acute{e}t}}(X)\to\mathrm{Gal}(\bar{k}/k)\to
1$ is exact. It also proves the analogous results for the `groupe fondamental
\'elargi' of SGA3.
