I am a bit confused about a notation I am seeing while reading a paper. So basically I have a continuous map between topological spaces $f:X\to Y$, and I know that this induces a map on cohomology $f^*: H^*(Y;R)\to H^*(X;R)$ for some chosen coefficient ring $R$. But the author of the paper I am reading tends to write $f^*:H^*(Y;R)\to H^*(X;f^*R)$ and I am not sure what to make of this (is this the same map $f^*$ that I know??) What does $f^*R$ exactly mean? I know that this is not a typographical error because there are more than one occasions where the author uses this. Could someone please explain this map to me? Is this notation usual? I haven't seen them so far in algebraic topology textbooks that I read.
Edit: As requested, I cite the paper (version 1): https://arxiv.org/abs/2009.06023. See Proposition 7.3 and 7.4.