Final answer:
A protein with both an ER sorting signal and a nuclear localization signal is first processed in the endoplasmic reticulum and then transported to the nucleus, which is its final destination.
Step-by-step explanation:
If a protein has both an ER sorting signal and a nuclear localization signal, the final destination of that protein would typically be the nucleus. Proteins are firstly synthesized in the cytoplasm, often on the ribosomes of the rough endoplasmic reticulum (ER). The ER sorting signal directs these proteins to the ER for initial processing. Afterwards, proteins usually move to the Golgi apparatus for further sorting, packaging, and tagging. However, as they are synthesized, proteins with a nuclear localization signal are ultimately recognized and transported into the nucleus, as this signal has a high-importance sorting role. Hence, the initial ER sorting merely facilitates early protein folding and quality control before the nuclear localization signal takes precedence, guiding the protein to its final destination, the nucleus.