Summary: We take another approach to the well known theorem of Korovkin, in the following situation: \(X\),~\(Y\) are compact Hausdorff spaces, \(M\)~is a unital subspace of the Banach space \(C(X)\) (respectively, \(C_{{\mathbb R}}(X)\)) of all complex-valued (resp., real-valued) continuous functions on~\(X\), \(S\subset M\) a complex (resp., real) function space on~\(X\), \(\{\phi_{n}\}\)~a~sequence of unital linear contractions from \(M\) into~\(C(Y)\) (resp., \(C_{{\mathbb R}}(Y)\)), and \(\phi_{\infty}\) a linear isometry from \(M\) into~\(C(Y)\) (resp., \(C_{{\mathbb R}}(Y)\)). We show, under the assumption that \({\varPi}_{N} \subset {\varPi}_{T}\), where \({\varPi}_{N}\) is the Choquet boundary for \(N = \text{Span} (\bigcup_{1\leq n\leq \infty}N_n)\), \(N_n=\phi_{n}(M)\) (\(n=1,2,\ldots,\infty\)), and \({\varPi}_{T}\) the Choquet boundary for \(T=\phi_{\infty}(S)\), that \(\{\phi_{n}(f)\}\) converges pointwise to \(\phi_{\infty}(f)\) for any \(f\in M\) provided \(\{\phi_{n}(f)\}\) converges pointwise to \(\phi_{\infty}(f)\) for any \(f\in S\); that \(\{\phi_{n}(f)\}\) converges uniformly on any compact subset of~\({\varPi}_N\) to \(\phi_{\infty}(f)\) for any \(f\in M\) provided \(\{\phi_{n}(f)\}\) converges uniformly to \(\phi_{\infty}(f)\) for any \(f\in S\); and that, in the case where \(S\) is a function algebra, the \(\{\phi_n\}\) norm converges to~\(\phi_{\infty}\) on~\(M\) provided the \(\{\phi_{n}(f)\}\) norm converges to~\(\phi_{\infty}\) on~\(S\). The proofs are in the spirit of the original one for the theorem of Korovkin.