A recently proved identity due to Duminil-Copin and Smirnov connects different generating functions for self-avoiding walks on the honeycomb lattice. Their identity holds only at the critical step fugacity $x_c,$ and was used by them to prove that $x_c=1/(1+\sqrt{2}).$ We extend their identity off-criticality, allowing us to prove certain exponent inequalities, and to prove an identity connecting the critical exponent describing the winding-angle of SAW with the exponents for SAW in a half-plane. Other extensions are also mentioned.