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To start with think of $Cat$ as a 1-category. The functor $Obj:Cat \to Set$ sending a small category to its set of objects is a fibration. This can be easily seen by constructing, given a category $C = (C_1 \rightrightarrows C_0)$ and a function $f:A \to C_0$, the set of arrows $A^2 \times_{f,C_0^2}C_1$ (the pullback of $(s,t):C_1 \to C_0^2$) of the category $C[f]$.

The cartesian lift of $f$ is then the canonical functor $F:C[f]\to C$.

Now given another function $g:A\to C_0$ -- giving rise to $G:C[g]\to C$ -- and a natural transformation $F \Rightarrow G$ there is a canonical isomorphism $C[f]\simeq C[g]$ over $C$. Thus if we think of Cat as a 2-category, there is something extra going on. For example, one gets a pseudofunctor $Set \to 2Cat$ on choosing specified pullbacks to define $C[f]$.

My question(s):

Has this phenomenon been studied before? (I would think so) Does this make $Obj$ a fibration of 2-categories (see e.g. Hermida, or Bakovic)? Or is this a more 'classical' concept? More basically, where was this fact first pointed out?

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