Question

$f_*$ and $f^*$ for direct and inverse image. We really should use this right from the beginning, for functions $f\colon X\to Y$, where $f_*\colon P(X)\to P(Y)$ ($P(X)$ being the power set) and $f^*\colon P(Y) \to P(X)$ instead of the awful notations $f(A)$ and $f^{-1}(B)$ for subsets $A$ of $X$ and $B$ of $Y$.