Finite-dimensional representations of Yangians in complex rank

We classify the "finite-dimensional" irreducible representations of the Yangians $Y(\mathfrak{gl}_t)$ and $Y(\mathfrak{sl}_t)$. These are associative ind-algebras in the Deligne category $\textbf{Rep}(GL_t)$, which generalize the regular Yangians $Y(\mathfrak{gl}_n)$ and $Y(\mathfrak{sl}_n)$ to complex rank. They were first defined by Etingof in his paper "Representation theory in complex rank". Here we solve Problem 7.2 from this paper. We work with the Deligne category $\textbf{Rep}(GL_t)$ using the ultraproduct approach introduced by Deligne and discussed by Nate Harman in arXiv:1601.03426.