The category $C_{F}$ possesses a weak initial object $I_{F}$, i.e. an object that is unique up to not necessarily unique isomorphism.
Let $F$ be a field and $L$ be its minimal subfield (the smallest subfield contained in $F$). Then either $L=\mathbb{F}_{p}$ for some prime $p$ or $L=\mathbb{Q}$.
Assume first that $F$ is of finite type over $L$. Let $n\in\mathbb{N}$ be the smallest natural number so that $F=L[x_{1},...,x_{n}]/\mathfrak{m}$ for some maximal ideal $\mathfrak{m}\subseteq L[x_{1},...,x_{n}]$. Let $\overline{x}_{i}$ be the image of $x_{i}\in L[x_{1},...,x_{n}]$ in $F$.
Let $\zeta:R\longrightarrow F$ be a surjection where $R$ is a local ring. Since every $\overline{x}_{i}$ has a (not necessarily unique) preimage $\zeta^{-1}(\overline{x}_{i})\in R$, there is a (not necessarily unique) morphism $\kappa:\mathbb{Z}[x_{1},...,x_{n}]\longrightarrow R$ that fits into a commutative diagram
$\require{AMScd}$
\begin{CD}
\mathbb{Z}[x_{1},...,x_{n}]@>{\kappa}>>R\\
@V{\pi}VV @VV{\zeta}V\\
L[x_{1},...,x_{n}] @>>{\chi}>F
\end{CD}
Let $\mathfrak{i}:=\chi^{-1}\pi^{-1}(0)=\pi^{-1}(\mathfrak{m})$. The ideal $\mathfrak{i}$ is always prime; it is maximal if and only if $L=\mathbb{F}_{p}$ for some prime $p$. Since $R$ is local, every element of $\mathbb{Z}[x_{1},...,x_{n}]$ is mapped by $\kappa$ onto something invertible in $R$. Hence $\kappa$ factors as
\begin{CD}
\mathbb{Z}[x_{1},...,x_{n}] @>>>\mathbb{Z}[x_{1},...,x_{n}]_{(\mathfrak{i})} @>{\lambda}>> R
\end{CD}
Thus $I_{F}:=\mathbb{Z}[x_{1},...,x_{n}]_{(\mathfrak{i})}$ is a weak initial object in the category $C_{F}$.
Note that the assignment $\kappa\longleftrightarrow\lambda$ is unique in both ways: To each choice of $\kappa$ there is a unique $\lambda$ and vice-versa.
Assume next that $F$ is of infinite type over $L$. Then $F$ is the direct limit of all morphisms $F'\longrightarrow F''$, where $F',F''$ are fields of finite type over $L$. Since the construction of $I_{-}$ is functorial and compatible with direct limits, $I_{F}$ can be defined as $I_{F}:=\lim_{F'\text{ of fin. t.}/L}I_{F'}$.
The initial object is strong, i.e. unique up to unique isomorphism, if and only if $F=L$.
Namely, if $F=L$, then $n=0$ and the unique morphism $\kappa:\mathbb{Z}\longrightarrow R$ induces a unique morphism $\lambda:\mathbb{Z}_{(\mathfrak{i})}\longrightarrow R$.
Else, if $F\neq L$, then $n\geq 1$ and for any $i\in\{1,...,n\}$ and any $s\in\mathfrak{i}\setminus\{0\}$, the map $\xi_{i,s}:x_{i}\mapsto x_{i}+s$ yields a nontrivial automorphism $I_{F}\longrightarrow I_{F}$ that commutes with the surjection $I_{F}\longrightarrow F$.
My guess is that the $\xi_{i,s}$ actually generate the whole group $\operatorname{Aut}(I_{F})$, but I have yet to figure out a proof for this...