By popular request from the last thread, this is a general dedicated to the discussion of ongoing study and work in homotopy theory. The /sci/ reading group is currently working on chapters 1 and 2 of Strom's Modern Classical Homotopy Theory.

Strom's book
>https://bookstore.ams.org/gsm-127
Chapter 1 exercises
>https://gofile.io/d/zeuzvm

Thread Prompt (Strom, Problem 1.77): Let $L\colon \mathcal{C}\to\mathcal{D}$ and $R\colon \mathcal{D}\to\mathcal{C}$ be a pair of adjoint functors. Show that there are natural maps $X\to RLX$ and $LRX\to X$ and that $RX$ is naturally a retract of $RLRX$.