I'm really confused about example 3.3.1 in https://web.ma.utexas.edu/users/sraskin/k.pdf.

The kernel $\ker(k[t]Mod \to k[t, t^{-1}]Mod)$ should be pairs $(V, T)$ where $V$ is a vector space and $T$ is a non-invertible endomorphism, but the claim is that it is $\mathcal{C}^{T}$ which is constructed in the paragraph before and has no non-invertible assumption on the endomorphism. The Ind completion should be doing something fancy to make this work but I don't know what