Usually I see the universal (small) cocartesian fibration $\mathcal{Z} \to Cat\_\infty$ constructed in a relatively opaque fashion, such as via the unstraightening construction.

I've stumbled on what seems to be a relatively simple construction, and am looking for a sanity check that I'm not overlooking something:

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> **Question:** Is $\mathcal{Z}$ equivalent to the full subcategory of $(Cat\_\infty)^{[1]}$ spanned by the arrows of the form $\mathcal{C}\_{/X} \to \mathcal{C}$, with the structure map is induced by the target operation $(Cat\_\infty)^{[1]} \to Cat\_\infty$?
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The rationale is as follows. For each small category, we have a series of fully faithful maps

$$ \mathcal{C} \subseteq \mathcal{P}(\mathcal{C})
\simeq RFib(\mathcal{C}) \subseteq (Cat\_\infty)\_{/\mathcal{C}} $$

with the first two maps being natural in $\mathcal{C}$.

The composite inclusion, I believe, sends an object $X$ to the right fibration $\mathcal{C}\_{/X} \to \mathcal{C}$.

The mapping $\mathcal{C} \mapsto (Cat\_\infty)\_{/\mathcal{C}}$ should classify the target map $(Cat\_\infty)^{[1]} \to Cat\_\infty$. While $RFib(\mathcal{C}) \subseteq (Cat\_\infty)\_{/\mathcal{C}} $ is not natural, it nonetheless seems very plausible that $\mathcal{C} \mapsto RFib(\mathcal{C})$ should be be the classifying map for the fibration $RFib \to Cat\_\infty$, where $RFib \subseteq (Cat\_\infty)^{[1]}$ is the full subcategory spanned by right fibrations.