Adversarial mindset means having an understanding of how an adversary thinks, what information is valuable to them, what sort of considerations are important to them, what kind of resources they have access to, and what sort of tactics they are likely to employ.
These elements allow your team to build an adversary profile that will act as a reference point throughout the operation. Having an understanding of an adversary's motivations is paramount. It is essential to understand what is desirable and of value to a particular adversary and what is most important to them.
For example, is the most crucial thing in targeting an organization not getting caught (as may be the case with corporate entities)? Does the adversary need to protect their reputation (such as often the case with state-sponsored)? Is it getting a return on their investment (such is often the case with organized crime)? Is it just about doing damage regardless of other costs (e.g., ideologically motivated actors)?
Understanding what is driving an adversary is going to help your team be able to replicate their approach and consequently most effectively defend against such attacks. This is the first place to start in terms of building an adversarial mindset.