Let $k$ be a field which is complete with respect to a non-trivial non-archimedean rank-1 valuation, and let $X$ be scheme which is locally of finite type over $k$. In section of 3.5 of [Berkovich's book](https://books.google.com/books/about/Spectral_Theory_and_Analytic_Geometry_Ov.html?id=6x5PIl2-DkkC), he defines the analytification of $X$ to be the closed $k$-analytic space $X^{\textrm{an}}$ representing the following functor: let $\textrm{An}\_k$ be the category of good analytic spaces over $k$, and let $F\_X \colon \textrm{An}\_k \to (\textrm{Sets})$ be the functor which sends a good analytic space $Z$ over $k$ to the set of morphisms $Z \to X$ of locally $k$-ringed spaces. 


In order for Berkovich's proof to work, it seems that we need the functor from $\textrm{An}\_k$ to the category of locally ringed spaces to be fully faithful; that is, every morphism of locally ringed spaces between two good analytic spaces must in fact be a morphism of analytic spaces over $k$. This fact is stated without proof on page 16 of Berkovich's [Trieste notes](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf), but how could one prove this?


However, there is also Remark 2.1.13 of Berkovich's book, which gives an example of a morphism (of locally ringed spaces) between $k$-affinoid spaces which is not a morphism of $k$-affinoid spaces. As $k$-affinoid spaces are examples of good analytic spaces, this seems to contradict the above assertion of fully faithfulness (granted, this particular example is over a trivially-valued field). This issue also does not seem to be addressed in Berkovich's [IHES paper](http://www.wisdom.weizmann.ac.il/~vova/IHES_1993_78_etale.pdf). 



Good analytic spaces over a field into locally ringed spaces is fully faithful
First, I would like to say that I do not understand why you need the full faithfullness of the analytification functor. It seems to me that the main point is to prove that giving a morphism from an analytic space $X$ to the affine analytic space of dimension $n$ is equivalent to giving $n$ global section of $X$.


Second, as you rightly say, the functor is not fully faithful over trivially valued fields nor for non-strict spaces (see Remark 2.1.13 of Berkovich's book), so let us assume that the valuation is non-trivial and that the spaces are strict. We can reduce to the case of a morphism between strictly $k$-affinoid spaces, say $f : X \to Y$. By taking global sections, you find an induced morphism between affinoid algebras $A\_Y \to A\_X$ and [BGR, Theorem 6.3.1/1] tells you that such a morphism is always continuous. Arguing as for schemes, you then prove that the induced morphism $X = \mathcal{M}(A\_X) \to \mathcal{M}(A\_Y) = Y$ coincides with $f$ for rigid points. By density of rigid points (which holds over a non-trivially valued field), it coincides with $f$ everywhere.


