Redundant conclusion in Lawvere theory example

[kevinclancy]

In example 1.3.4.6, presenting the theory $$\mathbf{Vect}$$ of real vector spaces as a lawvere theory, the last sentence draws that conclusion that $$g^* = f^*$$ as maps $$X^I \to X^J$$. What it should conclude instead is that $g = f$ as maps $J \to I$.

Maybe I'm being stupid, but I'm not sure why we need to prove that this functor is faithful. The definition of Lawvere theories does not mention anything about faithfulness.