How can I use a custom type as key for a map in C++?

I suspect you need

bool operator<(const Foo& foo1) const;

Note the const after the arguments, this is to make "your" (the left-hand side in the comparison) object constant.

The reason only a single operator is needed is that it is enough to implement the required ordering. To answer the abstract question "does a have to come before b?" it is enough to know whether a is less than b.

It's probably looking for const member operators (whatever the correct name is). This works (note const):

bool operator<(const Foo& foo1) const { return foo_value < foo1.foo_value;}

EDIT: deleted operator> from my answer as it was not needed (copy/paste from question) but it was attracting comments :)

Note: I'm 100% sure that you need that const because I compiled the example.