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As for the first question: Let us define a pointed scheme be a pair $(X,x)$, consisting of a scheme together with some point $x \in X$. Morphisms of pointed schemes are defined in an obvious way. Thus we get the category of pointed schemes. Also, we have the category of local rings with local ring homomorphisms. Then we have a functor

$\mathrm{Spec} : (\text{local rings})^{op} \longrightarrow (\text{pointed schemes})$

which maps a local ring $(A,\mathfrak{m})$ to the pointed scheme $(\mathrm{Spec}(A),\mathfrak{m})$. In the other direction, we have a functor

$\mathrm{Stalk} : (\text{pointed schemes}) \to (\text{local rings})^{op}$

which maps $(X,x) \mapsto \mathcal{O}_{X,x}$. Now Proposition 2.4.4. in EGA I may be reformulated as:

Proposition: $\mathrm{Spec}$ is left adjoint to $\mathrm{Stalk}$.

The counit of this adjunction is the canonical morphism $i : \mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ for a pointed scheme $(X,x)$ and the unit is the isomorphism $\mathcal{O}_{\mathrm{Spec}(A),\mathfrak{m}} \cong A_{\mathfrak{m}} \cong A$ for a local ring $(A,\mathfrak{m})$. So in more down-to-earth terms: $i$ is the universal morphism from the spectrum of a local ring to $X$ which maps to the closed point to $x$.

I doubt that for a pointed locally ringed space we have a morphism at all $\mathrm{Spec}(\mathcal{O}_{X,x}) \to X$. The reason is that the stalks of the structure sheaf are not sufficiently tied up together: There is no way of getting from prime ideals of the single local ring $\mathcal{O}_{X,x}$ to other points of $X$. You might try a topological (smooth) manifold with its sheaf of continuous (smooth) functions $(X,\mathcal{O}_X)$; see also this recent MO discussion about prime ideals in $\mathcal{O}_{X,x}$ in this example.

2 added 237 characters in body

As for the first question: Let us define a pointed scheme be a pair $(X,x)$, consisting of a scheme together with some point $x \in X$. Morphisms of pointed schemes are defined in an obvious way. Thus we get the category of pointed schemes. Also, we have the category of local rings with local ring homomorphisms. Then we have a functor

$\mathrm{Spec} : (\text{local rings})^{op} \longrightarrow (\text{pointed schemes})$

which maps a local ring $(A,\mathfrak{m})$ to the pointed scheme $(\mathrm{Spec}(A),\mathfrak{m})$. In the other direction, we have a functor

$\mathrm{Stalk} : (\text{pointed schemes}) \to (\text{local rings})^{op}$

which maps $(X,x) \mapsto \mathcal{O}_{X,x}$. Now Proposition 2.4.4. in EGA I may be reformulated as:

Proposition: $\mathrm{Spec}$ is left adjoint to $\mathrm{Stalk}$.

The counit of this adjunction is the canonical morphism $i : \mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ for a pointed scheme $(X,x)$ and the unit is the isomorphism $\mathcal{O}_{\mathrm{Spec}(A),\mathfrak{m}} \cong A_{\mathfrak{m}} \cong A$ for a local ring $(A,\mathfrak{m})$. So in more down-to-earth terms: $i$ is the universal morphism from the spectrum of a local ring to $X$ which maps to the closed point to $x$.

I doubt that for a pointed locally ringed space we have a morphism at all $\mathrm{Spec}(\mathcal{O}_{X,x}) \to X$. The reason is that the stalks of the structure sheaf are not sufficiently tied up together: There is no way of getting from prime ideals of the single local ring $\mathcal{O}_{X,x}$ to other points of $X$. You might try a topological (smooth) manifold with its sheaf of continuous (smooth) functions $(X,\mathcal{O}_X)$; see also this recent MO discussion about prime ideals in $\mathcal{O}_{X,x}$ in this example.

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As for the first question: Let us define a pointed scheme be a pair $(X,x)$, consisting of a scheme together with some point $x \in X$. Morphisms of pointed schemes are defined in an obvious way. Thus we get the category of pointed schemes. Also, we have the category of local rings with local ring homomorphisms. Then we have a functor

$\mathrm{Spec} : (\text{local rings})^{op} \longrightarrow (\text{pointed schemes})$

which maps a local ring $(A,\mathfrak{m})$ to the pointed scheme $(\mathrm{Spec}(A),\mathfrak{m})$. In the other direction, we have a functor

$\mathrm{Stalk} : (\text{pointed schemes}) \to (\text{local rings})^{op}$

which maps $(X,x) \mapsto \mathcal{O}_{X,x}$. Now Proposition 2.4.4. in EGA I may be reformulated as:

Proposition: $\mathrm{Spec}$ is left adjoint to $\mathrm{Stalk}$.

The counit of this adjunction is the canonical morphism $i : \mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ for a pointed scheme $(X,x)$ and the unit is the isomorphism $\mathcal{O}_{\mathrm{Spec}(A),\mathfrak{m}} \cong A_{\mathfrak{m}} \cong A$ for a local ring $(A,\mathfrak{m})$. So in more down-to-earth terms: $i$ is the universal morphism from the spectrum of a local ring to $X$ which maps to the closed point to $x$.

I doubt that for a pointed locally ringed space we have a morphism at all $\mathrm{Spec}(\mathcal{O}_{X,x}) \to X$. The reason is that the stalks of the structure sheaf are not sufficiently tied up together: There is no way of getting from prime ideals of the single local ring $\mathcal{O}_{X,x}$ to other points of $X$.