Artin representation of the braid group is well-defined

Corollary 1.8.3 (well-definedness part). Artin's assignment of an automorphism of the free group $F_n = \langle x_1,\dots,x_n\rangle$ to each braid generator extends to a group homomorphism $\xi : B_n \to \operatorname{Aut}(F_n)$, where the generator $\sigma_{i+1}$ acts by (1-14) of Birman p. 25: $x_i \mapsto x_i x_{i+1} x_i^{-1}$, $x_{i+1} \mapsto x_i$, and $x_j \mapsto x_j$ otherwise. Two ingredients are needed: each such endomorphism is invertible (it is an automorphism), and the assignment respects the braid relations (1-1) and (1-2). The assertion is the existence of the homomorphism $\xi$ with those values on the generators.