The next next paragraph is from Qing Liu's "Algebraic Geometry and Arithmetic Curves" p. 61. The definition of $\tilde f$ is unclear to me.
Here's the meaning of the notations. $k$ is a field, and $X$ is an algebraic variety over $k$. $\mathcal O_X$ is the sheaf given on $X$. $X^0$ is the topological subspace made up of the closed points of $X$. $k(x)$ is the residue field $\mathcal O_{X,x}/\mathfrak m_x$, where $\mathcal O_{X,x}$ is the stalk of $\mathcal O_X$ at $x$ and $\mathfrak m_x$ is the maximal ideal of $\mathcal O_{X,x}$.
Let's fix $x\in X^0$. Since $k(x)$ is an algebraic extension of $k$, there is an injective ring homomorphism $\varphi: k\to\bar k$. So $\tilde f(x) = \varphi(\text{the image of $f_x$ in $k(x)$})$. But what if there are two different injective ring homomorphisms $\varphi_1, \varphi_2: k\to\bar k$? How is $\tilde f(x)$ defined?