relation between pushforward and pullback

Question

I think its true that for an isomorphism $f: X\rightarrow Y$ of schemes there is an equivalence $\mathbf{Coh}(X) \leftrightarrow \mathbf{Coh}(Y)$ induced by $f_*$ and $f^*$. I'd be interested in how someone thinks intuitively about this and if that implies that $f^*=f^{-1}_*$? This seems to follow from this, but I've rarely seen it anywhere so far.

Answer 1

First, you are talking about categories, so instead of bijection you should talk about equivalence. Second, you don't have either of these in general; for instance if $X = \mathbb{P}^1$ and $Y$ is a point, $$ f_*(\mathcal{O}(-1)) = 0. $$