I am watching a youtube series on tensors called Tensors for Beginners by eigenchris, and while I understand how things like backward and forward transformations for matrices, linear maps, basis vector, covectors, and components are derived, I don't understand the system that is being used to label indices. Take for example the steps to derive the backward transformation for a linear map (and forgive me for not formatting this, but I have no idea how to write all these symbols in text): See here
What do the top and bottom indices mean? Why is the forward transformation for the linear map labeled with an l on the top, and an i on the bottom, but the backward transformation is labeled with an s on top and a t at the bottom. And then why is the original linear map labeled with a k and a j? Then how come when we cancel out the forward and backward transformations using the Kronecker delta, the opposite transformation being paired with the one in the original equation is labeled using different indices. What do these new indices represent? Several indices of new letters are added throughout the steps, but I don't get the logic behind how they are added and what they mean, and it's confusing me.