Short answer, no. Most people would agree that it is harder to compromise than it is to cooperate; and, that people are far more willing to do the latter as well. I will even go as far as saying that, in my humble opinion, the lack of cooperation in a relationship tends to lead to needing to compromise; or worse, a serious break down in a relationship.
Let's start with the most generic definitions of the two in comparison. According the the Merriam-Webster Dictionary, the definition of Cooperate is: 1) to act or work with another or others: act together or in compliance 2) to associate with another or others for mutual benefit.
The same source defines Compromise as: 1a) settlement of differences by arbitration or by consent reached by mutual concessions [side note, the second option there is usually the goal reached by mediation] 1b) something intermediate between or blending qualities of two different things 2) a concession to something derogatory or prejudicial. In the definition of Compromise, we see the word concession a few times; let's define that as well. The definition of Concession, per the same source, is: [side note, we will limit the definition to the relevant meaning, as there is an unrelated meaning] 1a) the act or an instance of conceding (as by granting something as a right, accepting something as true, or acknowledging defeat) 1b) the admitting of a point claimed in argument 2) something conceded [acknowledged, relinquished, accepted, or granted] or granted 2a) acknowledgment, admission 2b) something done or agreed to usually grudgingly in order to reach an agreement or improve a situation. Now, after having a thorough definition of both compromise and cooperation, you can see there is a stark difference.
Now that we know the differences, and to be fair, some similarities between the two, let's talk about each one in depth. We will start by discussing cooperation in relationships, and it's importance. Having a cooperative relationship is very important, because the alternative is a recipe for trouble. Imagine if you and your partner could not work together on even the simplest of tasks without bickering or giving up. If you are asking yourself if your current relationship is a cooperative one, ask yourself this: do you think it would be easy or difficult for you and your partner to work on a day long DIY project without arguing? How long does it take for you and your partner to agree on a movie or restaurant? Could you and your partner do well in a potato sac race if you had to share a sac? Are you and your partner good at playing games together like Pictionary or Charades? These are all examples of things that require cooperation to be successful at. Now, if your answers to those questions gives you pause about the future of your relationship, don't let it. There are many ways a relationship can grow in level of cooperation; knowing it is a problem and working on it is the first step to improvement. In my opinion, you cannot have a solid foundation in a relationship, in which to build on, if you struggle with balance. Balance and harmony are healthy results of cooperation; as opposed to lack of cooperation, which tends to lead to an imbalance and cause resentment on one side or both. Working in harmony helps strengthen the bond between partners, and maintaining a healthy level of cooperation should be a consistent goal. We tend to see high levels of cooperation in a relationship where each individual focuses on joint motives rather than their own motives. Empathy and navigating disagreements well are positive signs of cooperation in a relationship. Some people think a couple who has polar opposite personalities cannot work out, but that is not the case. Studies find that commonalities in personalities are less vital to a relationship than the ability to cooperate with one another. In fact, most people who think they are the polar opposite of their partner don't realize they are more similar than what shows on the surface. Most people magnetize to others who share the same core values, and have common goals or visions of their future; and, that matters more than being into the same music or being an extrovert vs. introvert.
Next, let's talk about Compromise, and how it factors in on relationships. In my opinion, compromising is something that comes into play once there is an imbalance of some kind, and the level of cooperation has either failed to exist or has somehow tilted more to one side. At this point, it will be clear that one party is either not aware of the needs/wants of the other, or just does not want to do the thing/s the other needs/wants. This is where negotiating begins, which can lead to a compromise of some kind. This will usually mean someone, if not both, will agree to doing something or giving something up they were not planning on. This is sometimes an issue when one or both parties to a relationship are focused more on ego than cooperation; or, when there are serious differences in opinion on whether or not something is necessary, important, or even healthy for the relationship or individual. Other times, cooperation is avoided all together, as one party is hopeful that the other party will let go of some wants/needs that don't align with theirs (ego). Compromising can be a very healthy and necessary tool in a relationship, especially where conflict has come into play; but it can also be damaging. An example of a compromise being unhealthy in a relationship is when the compromise leads a person to agree to something they fundamentally disagree with or feel will be sacrificing too much of their own wants/needs. An actual example would be this:
Tim and Kate meet. Tim has three children from a previous marriage, and made it very clear he does not want any more children. Kate is fine with this, because kids weren't really at the forefront of her mind, and she enjoyed her freedom to travel (a mutual passion they share). Tim and Kate get married three years later. On their ten year anniversary, Kate tells Tim she is having a change of heart on the kid subject, and now feels like it is something she finds herself desiring. Tim's children are nearly all off to college, and both Tim and Kate had always planned to retire early, travel the world, and live life as global nomads; both agreed to this and their plans had always moved in the direction of achieving that goal/lifestyle. Tim has not had a change of heart, does not want to start the cycle of raising a child again, and has worked hard his entire life to achieve his goals, which are just a few years away. In this scenario, either a life/future altering compromise must be made, which could be very damaging to one party's happiness and/or self actualization; or, they may not feel their futures are capable of aligning in a healthy way and choose to end their marriage. In this case, perhaps the issue at hand is one that is so serious and such a core part of who they are or what they want their lives to be, that perhaps compromising is not the best option. If Kate gives in, she will never mother her own children, which she now yearns to do; and, if Tim gives in, he is locked into raising another child full-time for at minimum 18 years, which was not factored into his future plans or budget. Both compromises come at a large cost/sacrifice. At times, when compromises ask too large of a cost/sacrifice it is almost inevitable to cause serious issues in the long run, and build a high level of resentment.
When a compromise is healthy, it eventually gets easier for the reluctant party, because ultimately the reward of having a happy partner and more balanced relationship, will outweigh the sacrifice or change made. When one chooses to compromise in a way that makes the other happy, but leaves them miserable, the sacrifice is bound to be an unnecessary effort, as the relationship will start to deteriorate. Balance is absolutely vital in a healthy relationship; and, that looks very different for every couple. But, each individual knows what their deal-breakers are; and, when they compromise with key ones, it will not work out, period. Nobody should be asked to compromise their own livelihood and happiness; and, to expect or want that is toxic behavior/thinking. Having said that, could Tim and Kate work things out? Sure! But, only Tim and Kate know if there is a compromise that could be made that could result in them both being happy. Like Kate has had a complete shift in her thinking of parenting another child from birth, and changing her initial future plans, perhaps Tim may find a way to fit that into his life projection without having to sacrifice his core wants/needs.
The bottom line is, the two are very different. And, if you work hard at being consistent at cooperation and aiming for mutually beneficial goals, then you will find yourself having to compromise less. Being a professional mediator, I know the importance of effectively doing both in a healthy partnership. This is especially true in mediation, where, in the majority of cases, there has been a break down in cooperation in a relationship (regardless as to whether it is a business partnership, intimate relationship, neighbor, academic, etc.), and the parties are now in a deep state of conflict, to where compromise is the only possible option in reaching a mutual agreement. In the mediation setting, when compromise is the only option (outside of an impasse), the win-win result of an agreement is almost never going to result in both parties being completely satisfied with the result. But, most people are content with having settled the conflict. At times, the lack of cooperation in a relationship is so extreme that compromise cannot be an attainable goal; and, that is when you must have an outside source (usually a judge) make the decision as to the outcome of the conflict.