Summary: Needed narrowing is a complete and optimal operational principle for modern declarative languages which integrate the best features of lazy functional and logic programming. We investigate the formal relation between needed narrowing and another (not so lazy) narrowing strategy which is the basis for popular implementations of lazy functional logic languages. We demonstrate that needed narrowing and lazy narrowing are computationally equivalent over the class of uniform programs. We also introduce a complete refinement of lazy narrowing, called uniform lazy narrowing, which is still equivalent to needed narrowing over the aforementioned class. Since actual implementations of functional logic languages are based on the transformation of the original program into a uniform one -- which is then executed using a lazy narrowing strategy -- our results can be thought of as a formal basis for the correctness of these implementations.