Im reading Vistolis "Notes on Grothendieck topologies, fibered categories and descent theory", and when i read the definition of a fibered category, it struck me as very similiar to the definition of a slice category.
I think the category of fibered categories $\textbf{fCat}$ over $\mathbb{E}$ is equivalent to the category $\textbf{Cat}/\mathbb{E}$ through $T: \textbf{Cat}/\mathbb{E} \to \textbf{fCat}$ where $T(f:X \to \mathbb{E})=X$ and the identity on morphisms and $K: \textbf{fCat} \to \textbf{Cat}/\mathbb{E}$ which takes $X$ and sends it to its equipped functor $p: X \to \mathbb{E}$.
Im i on the right page?
Being fairly sure of the above i conjectured if one could write the category of fibered categories over $\mathbb{E}$ as a cone category. i.e. if there exists a functor of a specific shape $I$, say $F: I \to \textbf{Cat}$ such that $\textbf{Cone}(F)$ is a fibered category. The answer is yes and here why i think so! It is a subcategory of $\textbf{Fun}(I,\textbf{Cat}) / F$, which (i think) is fibered.
Note that $F \in \text{ob}\textbf{Fun}(I,\textbf{Cat})$. For every object $C \in \text{ob} \textbf{Cat}$ there is a constant functor $\Delta C \in \text{ob}\textbf{Fun}(I,\textbf{Cat})$, which on objects $i \in \text{ob} I$ is given by $i \mapsto C$ and on morphisms $\varphi \in \text{mor} I$ is given by $\varphi \mapsto \text{id}_C$.
Natural transformations $\Delta C \Rightarrow F$ are, by definition, exactly cones from $C$ to $F$, which means every object of $\textbf{Cone}(F)$ is an object of the slice category $\textbf{Fun}(I,\textbf{Cat}) / F$. Now $\Delta$ extends to the faithful functor $\Delta(-): \textbf{Cat} \to \textbf{Fun}(I,\textbf{Cat})$. If $f \in \textbf{Hom}_\textbf{Cat}(C,D)$, then one defines $\Delta(f): \Delta C \to \Delta D$ by $(\Delta f)_{i \in \text{ob} I}=f$. Given two cones $\gamma: \Delta C \to F$ and $\delta: \Delta D \to F$, and $p \in \textbf{Hom}_\textbf{Cat}(C,D)$ then $\Delta(p): \Delta C \to \Delta D$ is a morphism in $\textbf{Fun}(I,\textbf{Cat})/F$. Which, since $\Delta$ was faithful, means $\textbf{Cone}(F)$ is a subcategory of $\textbf{Fun}(I,\textbf{Cat}) / F$.
So i closing is the equivalence even correct? and are my ideas about why cone categories are fibered correct?