Let $\mathsf{Top} \downarrow X$ denote the slice category of topological spaces over some base space $X$. Then the functor $\Gamma: \mathsf{Top} \downarrow X \to \mathsf{Sh}(X)$ which assigns to any map of spaces $p: E \to Y$ the sheaf of sections has a left adjoint, which is the etale space (i.e. geometric realization) construction $\mathrm{Et}: \mathsf{Sh}(X) \to \mathsf{Top} \downarrow X$. The latter functor is fully faithful, and hence the unit $\eta: 1 \Rightarrow \Gamma \mathrm{Et}$ is a natural isomorphism. The counit $\varepsilon: \mathrm{Et} \Gamma \Rightarrow 1$ is not a natural isomorphism. However, by the triangle identities the whiskered transformations $\Gamma \varepsilon : \Gamma \mathrm{Et} \Gamma \Rightarrow \Gamma$ and $\varepsilon \mathrm{Et}: \mathrm{Et} \Gamma \mathrm{Et} \Rightarrow \mathrm{Et}$ are both natural isomorphisms.
For a continuous map $f: Y \to X$, often the inverse image of sheaves $f^{-1}: \mathsf{Sh}(Y) \to \mathsf{Sh}(X)$ is defined to be the composite $$f^{-1} := \Gamma f^* \mathrm{Et}$$ where $f^*: \mathsf{Top} \downarrow Y \to \mathsf{Top} \downarrow X$ is the pullback or base change.
The wikipedia page states that "the pullback of bundles then corresponds to the inverse image of sheaves." I would like to show more generally that the pullback between slice categories over topological spaces corresponds to the inverse image of sheaves. In other words, I would like to prove the following:
Proposition: $f^{-1} \Gamma \cong \Gamma f^*$.
Attempted proof: By definition, $f^{-1}\Gamma = \Gamma f^* \mathrm{Et} \Gamma .$ The (whiskered) counit gives us a natural transformation $$\Gamma f^* \varepsilon: f^{-1}\Gamma = \Gamma f^* \mathrm{Et} \Gamma \Rightarrow \Gamma f^* $$ How can I verify that this is a natural isomorphism? Or am I going about this the wrong way? I'd prefer an "abstract" proof, in this sort of spirit, if possible.
Nice question, but I think this is false even in the case of vector bundles.
Take for example $f:\{0\}\rightarrow\mathbb{R}$ the inclusion and $p:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ the first projection so we can see $\mathbb{R}\times\mathbb{R}$ as an object in $\mathrm{Top}/\mathbb{R}$.
We have $f^*p=\mathbb{R}$ seen as a toplogical space over the point, and $\Gamma f^*p=\mathbb{R}$ seen as a sheaf on the point.
On the other hand $\Gamma p$ is the sheaf of continuous function on $\mathbb{R}$, and $i^{-1}\Gamma p$ is the set of germs of continuous function at $\{0\}\subset\mathbb{R}$ (seen as a sheaf over the point).
Note that the natural transformation you defined is the evaluation map. But the latter is much larger than the former.