# Why adjunction appear to preserve stalk?

Question:

Let $$(f,f^{\flat}) \colon (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$$ be a morphism of locally ringed space. $$f \colon X \to Y$$ is a continuous map and $$f^{\flat} \colon \mathcal{O}_Y \to f_* \mathcal{O}_X$$ is a homomorphism of sheaves of rings.

Assume that $$x \in X$$. There are $$2$$ two ways.

[1] The maps $$f^{\flat }_U \colon \mathcal{O}_Y(U) \to \mathcal{O}_X(f^{-1}(U))$$ for every open neighborhood $$U$$ of $$f(x)$$ induce maps $$\mathcal{O}_Y(U) \to \mathcal{O}_{X,x}$$ and a map $$f_x^{\sharp} \colon \mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$$

[2] Let $$f^{\sharp} \colon f^{-1}\mathcal{O}_Y \to \mathcal{O}_X$$ be the morphism corresponding to $$f^{\flat}$$ by adjointness. Since $$(f^{-1}\mathcal{O}_Y)_x = \mathcal{O}_{Y,f(x)}$$, we get $$f_x^{\flat} \colon \mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$$ .

I know these $$2$$ maps coincide. It is possible to represent these maps by taking elements. But I believe there is a proof that is shorter, more suggestive and more category-theoretical.

Thanks.