Let $(X, \mathcal{O}_X), (Y, \mathcal{O}_Y)$ are ringed spaces. Let $\mathscr{F}$ is $\mathcal{O}_Y$-module and $f: X \to Y$ is morphism of ringed spaces.
I want to prove that symmetric power commuting with pullbacks. I already prove that pullbacks commuting with tensor powers. So, my plan is the following, I consider exact triple $$0 \to \mathscr{I} \to T^n \mathscr{F} \to S^n \mathscr{F} \to 0$$ and want to prove that triple $$0 \to f^* \mathscr{I} \to f^* T^n \mathscr{F} \to f^*S^n \mathscr{F} \to 0$$ are exact too. But $f^*$ apriori only right-exact and I don't see any reason why it is left exact in this particular case. Hope for your help!