Suppose I construct the variable $U_i = S_i + T_i $, and I want to estimate the equation
$$ Y_i = \alpha + \beta U_i + \gamma X_i + \epsilon _i $$
Suppose $X_i$ and $S_i$ are exogenous, but $T_i$ is endogenous.
A valid instrument for $X_i$ appears to be be $S_i$, so that the first stage estimation would be
$$ U_i = \alpha_U + \beta_U S_i + \gamma_U X_i + \eta_i $$
Since $S_i$ is used in the construction of $U_i$, it should be a relevant instrument, and since we assume that $S_i$ is exogenous, it should also satisfy the exclusion restriction.
Supposing that the above is correct, my question is whether the IV estimation of $\beta$ is the same as just estimating $\beta_S$ in the following equation:
$$Y_i = \alpha_S + \beta_S S_i + \gamma_S X_i + \xi_i $$
Thanks for your help!