Let $f \colon X \to Y$ be a continuous map and $\mathcal{O}_X$ a sheaf (of sets) on $X$.
Question: Is the stalk $(f_*\mathcal{O}_X)_{f(p)}$ for $P \in X$ isomorphic to the stalk $\mathcal{O}_{X,P}$ ? If it is, how exactly does the isomorphism map elements of these stalks (which are equivalence classes of pairs $(U,g)$) to one another ? Is the isomorphism unique/universal/canonical in some sense?
Remark: I know (just) a little bit about adjunctions, but if the answer is related to the inverse image functor (which I don't feel familiar with) and the adjunction to the pushforward, please give me a detailed explanation. Thank you.
No, there is no such isomorphism in general. For instance, let $X$ be a proper $k$-variety of positive dimension and $f:X\to Y=\textrm{pt}$ its structural morphism. Let $\mathcal O_X$ be the sheaf of regular functions on $X$. Then for any regular point (say) $P\in X$ the stalk $(f_\ast\mathcal O_X)_{f(P)}=(f_\ast\mathcal O_X)_\textrm{pt}=\mathcal O_X(X)=k$, but $\mathcal O_{X,P}$ is a ring of dimension $\dim X>0$.