Minor technical correction- it would be false to say that GI came from fungi as we humans have GI, but don't descend from fungi (we share a common ancestor) and likely haven't received the genes for GI (horizontal gene transfer) from them.

Personally I figure that once you have a complex enough system with feedback regulation and self-preservation, something like general intelligence occurs spontaneously through the normal mechanisms of evolution (probably more than once in different kingdoms) because it provides a heritable survival edge.