  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.
