This is my first post here, as someone from Mathsstack suggested this might me a more suitable forum for this specific question.

I have been reading some texts by Joaquim Lambek on formal languages, grammar and so on, and at some point he mentions that the kind of approach he advocates in the late 90s and 2000s has the advantage, as compared to some similar ones from the end 60s, that they allow for double adjoints, which, in turn, account for what other schools address with the use of traces.

Does that mean that double adjoints were ony developed in logic and mathematics in the 70s or later? Or how should I understand that claim? I am assuming that the authors in the 60s were not simply oblivious

I would be very grateful if you could me illustrate me on the history of double adjoints, with references, milestones, etc...

Thanks in advance.