On Guay's evaluation map for affine Yangians

We give a detailed proof of the existence of evaluation map for affine Yangians of type A to clarify that it needs an assumption on parameters. This map was first found by Guay but a proof of its well-definedness and the assumption have not been written down in the literature. We also determine the highest weights of evaluation modules defined as the pull-back of integrable highest weight modules of the affine Lie algebra ${\widehat{\mathfrak{g}\mathfrak{l}}}_N$ by the evaluation map.