I am a PhD student seeking to collaborate with PhD student/professors from other department. I have never made such an initiative before, and I lack the general idea on how to start. In particular:
If I already have some specific research questions in mind (which I will use for my dissertation research), is it important to ensure that the research lab is a reasonable match to my specific research goals? For example, if my field is natural language processing and I am specifically interested in doing a research in Transformer language models, should I ensure that the potential research lab for collaboration is also heavily into Transformer research? or can I try contacting any natural language processing labs in general, even if the lab's subarea of research is not a great match to my research goals?
My main supervisor is open to the idea of collaboration. To make the collaboration to happen, can I just contact the individual researcher/PhD student whom I am interested in working with? or should I first contact the professor who is leading the lab that the researcher/PhD student belongs to?
Should my main supervisor contact the research group on my behalf, instead of me approaching them?
If I collaborate with a research lab in different department, does this automatically mean that I will formally be a member of that research lab and have the professor who is leading the lab as my co-supervisor?
Is it okay for PhD student to collaborate with research labs from different university (assuming that my main supervisor is okay with this)?