adjudication

Could anyone help me understand in what meaning the word 'adjudication' is used in the following context?

"The disadvantage of customary law is that it is not based on a written agreement and, consequently, that it is not easy to determine to what extent a particular rule has attained customary status. In reality, State practice tends to be examined and customs identified by national and international courts and tribunals tasked with the interpretation and adjudication of international law. The ICRC’s extensive study on customary international humanitarian law is also a widely recognized source of reference in this respect".

Adjudication only has one principal meaning, and this is the one being used here. Is there something in particular troubling you?

I am afraid, my problem is that I think not in English, but in Russian, and I have problem of constructing a Russian sentence in which the words "adjudication of international law" makes sense. If it were 'adjudication on the basis of international law' nothing would trouble me. But why 'of'?

When you write that adjudication has only one one principal meaning, which meaning do you have in mind? 'Giving a decision on sth, especially a formal or binding one'? (Collins English Dictionary). If yes, I do not understand how a tribunal can 'give a formal or binding decision OF international law'. I can only imagine a tribunal giving a decision BASED ON international law.

Courts and tribunals are often required to make rulings on what the international law actually is/means. That is what is meant here.
For example an international law which applies to "health products" may well require an adjudication on whether a certain item is or is not a health product.

Thank you for this explanation.