OLS Estimator for linear regression model is a linear estimator

Question

Proofs for the Gauss-Markov Theorem often assume without showing that the OLS $\hat{\beta}=(X^TX)^{-1}X^TY$ is linear, that is that for $y_1,y_2 \in \mathbb{R}^n $ $$\hat{\beta}(y_1+y_2)=\hat{\beta}(y_1)+\hat{\beta}(y_2)$$ Is this just as trivial as putting $$\hat{\beta}(y_1+y_2)= (X^TX)^{-1}X^T(y_1+y_2)=...$$ I don't think so, I think there is more to it that I can't figure out and would appreciate your help.