Proof.
By proposition 7.4.11, a \(\mathbb T_2 \otimes \mathbb E_1\)-structure on \(F\) is a lift in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{({1}, {1})})\): 
Explicitly, the space of \(\mathbb T_2\otimes \mathbb E_1\)-structures on \(F\) is therefore the \(1\)-groupoid of weak lifts from theorem 2.6.4.([002J]). By theorem 2.6.4, this is equivalent to the set of prebraidings on \(F\) in the sense of definition 2.4.8. This completes the proof of part ([00FB]). Part ([00FC]) follows by applying statement ([00FB]) to the case \(F=\mathrm{id}_{\mathcal A}\). ◻