Sure, no problem.
Essentially, what has been done is that the equation
(u_h, \beta\cdot\mathbf{n} v_h)_{\partial\Omega_\_} = (g, \beta\cdot\mathbf{n} v_h)_{\partial\Omega_\_}
has been subtracted from the weak form, such that this condition holds when solving for u_h. Since on the inflow boundary \beta\cdot\mathbf{n} < 0, it is important that the term is subtracted, and not added, in order to maintain coercivity of the bilinear form.
Usually though, weak imposition of Dirichlet boundary conditions is done by including these terms scaled with a penalty parameter and a mesh-dependent metric (often the mesh cell diameter), to ensure that the terms scale properly with mesh refinement, ensuring stability of the scheme with respect to the discretization. See e.g. Weak imposition of Dirichlet conditions for the Poisson problem — FEniCSx tutorial and the reference therein to Nitsche’s method, which is a method for imposing Dirichlet boundary conditions weakly.