A frequent origin of the copula is a demonstrative or anaphoric pronoun. Consider the case of the Chinese copula (Li & Thompson 1977). In Archaic Chinese, nominal clauses contained no copula. The subject of a nominal predication, especially a relatively heavy one, could be topicalized by left-dislocation. This necessitates a substitute in the subject position of the nominal clause, a demonstrative or personal pronoun which anaphorically takes up the topicalized NP. The resulting nominal clause is, of course, syntactically completely unmarked. The complex sentence structure is as follows:
S[ NP S[ Dem NP ] ]
The Dem in Archaic Chinese is shì. By the 1. cent. AD, this construction was sufficiently grammaticalized to be reanalyzed as S[NP Dem NP]. Here shì already functions as a copula, one criterion being that it is indifferent as to the person of the subject. About the same time, it ceases to be used as a demonstrative, while in its copula function it becomes increasingly obligatory.
Similar developments are known from other languages. In Kalamang (Papuan, Greater West Bomberai), the distal demonstrative me has assumed functions of a thematic structure articulator and a copula (Visser 2022).
References
Li, Charles N. & Thompson, Sandra A. 1977, “A mechanism for the development of copula morphemes.” Li, Charles N. (ed.) 1977, Mechanisms of syntactic change. Austin & London: University of Texas Press; 419-444.
Visser, Eline 2022, A grammar of Kalamang. Berlin: Language Science Press (Comprehensive Grammar Library, 4). [online]