I can understand why the image of $\alpha\in \operatorname{Hom}_K(U,\operatorname{ind}_M^K(V_\sigma))$ lies in the subspace of continuous functions of $\operatorname{ind}_M^K(V_\sigma)$, and then the two books both perform evaluating at 1, but I don't understand its necessity, as the remaining paragraphs showing injectivity and surjectivity doesn't use the necessity assertion. The symbol $\mathcal{C}$ and $\operatorname{Hom}_K$ both denote the intertwining operators.