Try to take advantage of the stable nature of insertion sort

insertion_sort.hpp

template<typename BidirectionalIterator, typename Compare, typename Projection>
auto insertion_sort(BidirectionalIterator first, BidirectionalIterator last,
                    stable_compare<Compare, Projection> compare, utility::identity)
    -> void
{
    auto projection = compare.projection();
    cppsort::detail::insertion_sort(
        first, last, compare.compare(),
        [projection](auto&& value) mutable -> decltype(auto) {
            return projection(std::forward<decltype(value)>(value).get());
        }
    );
}