Relating Stated Skein Algebras and Internal Skein Algebras

We give an explicit correspondence between stated skein algebras, which are defined via relations on tangles in [Costantino F., L\^e T.T.Q., arXiv:1907.11400], and internal as endomorphism algebras free cocompletions of categories [Ben-Zvi D., Brochier A., Jordan J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or [Gunningham S., Safronov P., arXiv:1908.05233]. Stated surfaces with multiple boundary edges we generalise this context. Now, one needs to distinguish left right edges, explain phenomenon using a half-twist. prove excision properties multi-edges categories, agreeing when $\mathcal{V} = \mathcal{U}_{q^2}(\mathfrak{sl}_2)\text{-}{\rm mod}^{\rm fin}$. Our proofs mostly based theory do not require the reader be familiar formalism higher categories.