Well-pointed object in a category

An object $A$ in a category with a terminal object is well-pointed when every monomorphism $m:B\to A$ through which all points $1\to A$ factor is invertible. This definition concerns detection of proper subobjects by points, and does not designate a chosen basepoint. If there are no points, it asserts that every subobject is the whole object. In particular, the empty set is well-pointed.