I'm not clear on how you could have an inter-rater comparison that is not mutually exclusive. In particular, for any given coded text segment, you construct a table of agreement and disagreement. Either both coders gave out the code, or only one did.

For example, two coders need to analyze and code 10 text. The text can belong to category A or B or both.  How could it work if I'd like to calculate the inter-rater reliability with Cohen's kappa?

You calculate kappa for Code A and then for Code B and average the two.

Kinda need to know more to be able to answer this correctly. It sounds as if you may need further pilot tests to have proper operationalizations. Recommend these articles:

Brennan, R. L., & Prediger, D. J. (1981). Coefficient kappa: Some uses, misuses, and alternatives. Educational and Psychological Measurement, 41(3), 687-699.

Banerjee, M., Capozzoli, M., McSweeney, L., & Sinha, D. (1999). Beyond kappa: A review of interrater agreement measures. Canadian Journal of Statistics, 27(1), 3-23.

O’Leary, S., Lund, M., Ytre-Hauge, T. J., Holm, S. R., Naess, K., Dalland, L. N., & McPhail, S. M. (2014). Pitfalls in the use of kappa when interpreting agreement between multiple raters in reliability studies. Physiotherapy, 100(1), 27-35.

If a text can be A, B, or A+B, you should code of A and B, and calculate IC reliability for these two variables. You also should calculate a variable A+B (you can do so by simply adding A and B codes for both rates) and compere IC for the third variable as well. This way, you are not only assessing if raters agreed if categories A or B apply, but also if raters consistently coded the presence of A+B in a text.