One Premise I worked under for API development is "The Consumer of the API adapts to the API." For example if I call Yelp or Twitter API I have to adapt to it. There is no way they are going to modify it for my requirements.
This leads me to an issue that has recently come up with a vendor. A vendor solution needs to call our APIs for integration. They are asking that we build our APIs per their spec (and there are some issues around best practices in how their APIs are designed).
This means we have to create an adapter to transform our GET/POST APIs (http error codes, properties) to match their spec. So that there are no coding changes / mapping / adapting required on their end when they call our APIs.
When working with a SaaS solution (that needs to consume data from your systems) and for that vendor to help with adoption of their software, I would still think the concept "The consumer of the API adapts" would apply and they should have an adapter layer.
Looking for best practices / thoughts on this approach from a wider audience.