  We report on a method to certify a unitary operation with the help of source
and measurement apparatuses whose calibration throughout the certification
process needs not be trusted. As in the device-independent paradigm our
certification method relies on a Bell test, but it removes the need for high
detection efficiencies by including the single additional assumption that
non-detected events are independent of the measurement settings. The relevance
of the proposed method is demonstrated experimentally with the certification of
a quantum frequency converter. The experiment starts with the heralded creation
of a maximally entangled two-qubit state between a single $^{40}$Ca$^+$ ion and
a 854$\,$nm photon. Entanglement preserving frequency conversion to the telecom
band is then realized with a non-linear waveguide embedded in a Sagnac
interferometer. The resulting ion-telecom photon entangled state is
characterized by means of a Bell-CHSH test from which the quality of the
frequency conversion is quantified. We demonstrate the successful frequency
conversion with an average certified fidelity of $\geq 84\,\%$ and an
efficiency $\geq 3.1\times 10^{-6}$ at a confidence level of $99\,\%$. This
ensures the suitability of the converter for integration in quantum networks
from a trustful characterization procedure.
