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