I will try to explain.

lower_bound/upper_bound allways work on vetor/array sorted by an order determined by some comparator. A comparator is function (or something usable like a function) which returns consistent result while comparing elements. In particular it must allways be true that if a<b && b<c then a<c.

A lot of classes have a default, or call it natural comparator defined. For pair<T1,T2> that comparator compares the first element, and ties on second element.

If we use sort/lower_bound/upper_bound without giving a comparator as argument, that default comparator is used. You will get a compiler/linker error if none is defined.

So, as an example how to implement a comparator for pair<int,string> comparing second before first? I usually do as inline function objects, like:

auto myComp = [&](pair<int,string> e1, pair<int,string> e2) {
  if(e1.second!=e2.second)
    return e1.second<e2.second;
  else
    return e1.first<e2.first;
};

This myComp can now be used like a function name, ie call it or give it as an argument to other function calls. ie:

vector<pair<int,string>> data(n);
...
sort(data.begin(), data.end(), myComp);

After having sorted the vector we may use binary search to find entries:

auto it=lower_bound(data.begin(), data.end(), { INT_MIN, "someString" }, myComp);
if(it!=data.end())
  cout<<"found first string >=someString at index="<<distance(data.begin(), it)<<endl;
else
  cout<<"notfound >=someString"<<endl;