If $f$ is a vertical arrow? in a double category$C$, its tabulator is the terminal object in the category whose objects are objects$T$ of $C$ equipped with a 2-cell $\pi\colon id_T\to f$ and morphisms $\pi'\to\pi$ are horizontal arrows?$g\colon T'\to T$ such that the horizontal composition of the identity 2-cell on $h$ and the 2-cell $\pi$ yields $\pi'$.

References

Marco Grandis, Robert Pare, Limits in double categories. Cahiers Topologie Géom. Différentielle Catég. 40 (1999), no. 3, 162–220.

Marco Grandis, Higher Dimensional Categories. From Double to Multiple Categories, doi

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