Think about it like a normal pointer:
int * const i;
is a const pointer to a non-const int. You can change the int, but not the pointer.
int const * i;
is a non-const pointer to a const int. You can change the pointer but not the int.
Now, for unique_ptr, it's a question of whether the const goes inside or outside the <>. So:
std::unique_ptr<int> const u;
is like the first one. You can change the int, but not the pointer.
What you want is:
std::unique_ptr<int const> u;
You can change the pointer, but not the int. Or perhaps even:
std::unique_ptr<int const> const u;
Here you can't change the pointer or the int.
Notice how I always place the const on the right? This is a little uncommon, but is necessary when dealing with pointers. The const always applies to the thing immediately to its left, be that the * (pointer is const), or the int. See http://kuhllib.com/2012/01/17/continental-const-placement/ .
Writing const int, might lead you to thinking int const * is a const-pointer to a non-const int, which is wrong.
.cannot be overloaded.constis shallow. Anint* constalso allows you to modify the pointed to object.std::experimental::propagate_constif you want to propagate it.const.