The segre embedding takes two schemes $X, Y$ with $G_m$ and (I think) produces an embedding $X/G_m \times Y/G_m \hookrightarrow (X \times Y)/G_m$, with $G_m$ acting diagonally on $X \times Y$.
My confusion is that this seems like a "limit of colimit" to "colimit of limit" sort of map. Or at least, by the nature of the way things are pointing can't make use of the universal properties of these objects. I feel like the more natural thing to consider would be a map $$ (X \times Y)/G_m \to X/G_m \times Y/G_m $$
which naively would send $[(x, y)]$ (the equivalence class of $(x, y)$ to $([x], [y])$, which is well-defined on equivalence classes. Is this related to the Segre embedding at all? Is this even a well-defined map? Is there any reason why this latter map isn't discussed as much? Is there a group-actiony perspective on the Segre Embedding? Thanks!