Effectivity of Generalized Double $\infty$-Categories

We construct an adjunction between $m$-categories internal to $(\infty,n)$-categories, called $(n,m)$-double $\infty$-categories, and filtrations $A_0\to \dots\to A_m$ where for all $i<m$, $A_i$ is a $(n+i)$-category. We show that this adjunction induces an equivalence between $(n,m)$-double $\infty$-categories admitting enough companions and filtrations such that each morphism $A_i\to A_{i+1}$ is essentially surjective on cells of dimension lower than or equal to $i$. This result can be seen as a $(\infty,n)$-categorical generalization of the equivalence between internal groupoids and effective epimorphisms in the category of $\infty$-groupoids proven by Rezk and Lurie. In the case $n=0$, this recovers the characterization of flagged $m$-categories given by Ayala-Francis, and in the case $n=1$, it allows us to prove some conjectures concerning the square functor and its variants, stated by Gaitsgory-Rozenblyum in the appendix of their book on Derived Algebraic Geometry.