In section 4.1, chapter 4 of Pierre Deligne's paper *La conjecture de Weil : I* (french version, translation to English) he states:

On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a non singular analytic space and purely of dimension $n+1$. Let $D=\{z: |z|< 1 \}$ the unit disc and $D^*=D-\{0\}$, and let $f: X\rightarrow D$ a morphism of analytic spaces such that:

$f$ is proper

$f$ is smooth outside of a point $x$ of the special fiber $X_0=f^{-1}(0)$.

In $x$, $f$ has a non-degenerate quadratic point.

Let $t\neq 0$ in $D$ and $X_t=f^{-1}(t)$ "the" general fiber.

With the above data he associate some results for the cohomology groups of the fibers.

Then, in section 4.2. He says: There is an analog of (4.1) in abstract algebraic geometry. The disk $D$ is replaced by the spectrum of a henselian discrete valuation ring $A$ with an algebraically closed residue field. Let $S$ be the spectrum, $\eta$ its generic point (spectrum of the field of fractions of $A$), $s$ the closed point (spectrum of the residue field). The role of $t$ is played by the geometric generic point $\overline{\eta}$ (spectrum of the closure of the field of fractions of $A$).

I am wondering why he replaces $t$ by the geometric generic point $\overline{\eta}$? In any sense, there is a one to one correspondence between them?