"Duty assignment" is a phrase commonly used to refer to a task or job that someone is required to complete. However, there are several synonyms for this term that can be used to add variety to writing or conversation. Some of these synonyms include "responsibility," "obligation," "mission," "errand," "chore," and "duty roster." These synonyms can also be useful in different contexts. For example, "responsibility" may be a better choice for a job that requires significant decision-making or leadership, while "errand" might be used for a smaller, more routine task. Ultimately, the choice of synonym will depend on the context and desired tone.