I am a little confused by the word "assignment". The dictionary says it refers to a piece of work that is given to someone as part of their job. For example, Tom assigns something to me. In other words, I am going to do the assignment. When I communicate with Tom, which possessive pronoun should I use? The task is assigned by Tom; I think it is reasonable to say "Tom's assignment" or "his assignment". On the other hand, I am supposed to handle this task; I think it makes sense to say "my assignment". It beats me.
In contractual terms, what you have here is an example of a unilateral contract. Whereas in a bilateral contract (far more common than a unilateral contract) there is a promise for a promise; in a unilateral contract there is a promise in exchange for some completed action.
When law enforcement, for example, offers a reward for information that leads to the arrest and conviction of a man on the FBI's Most Wanted List, there is a unilateral contract: If you give us the information that leads to the arrest and conviction of the criminal (the completed performance), we'll pay you $10,000 (the reward). In other words, there is a promise for a completed performance.
Similarly, when a colleague (or boss) gives you an assignment, in effect he "promises" to "reward" you with a thank you, or a "Well done!", or "I appreciate that!", or perhaps "Good. Here's another assignment," but only after you complete the assignment. By the way, if the assignor is your boss, the "consideration" (the agreed-upon reward) is that you get to keep your job, and your boss gets a completed assignment, which is his reward.
Again, in legal terms, what you have here are an offeror (the one who is giving the assignment) and an offeree (the one accepting the assignment). They come to a meeting of the minds regarding
who initially possesses what assignment,
who then gives the assignment to whom to possess,
and who in turn gives the completed assignment to whom to possess.