This question is a bit related to the question To what extent is a scheme morphism determined by its topological map?.
Let $X, Y$ be separated geometrically reduced schemes over a field $k$. Whenever I have a morphism $X \to Y$, I get a morphism $X(k^{\text{alg}}) \to Y(k^{\text{alg}})$ between the $k^{\text{alg}}$-points, where $k^{\text{alg}}$ is the algebraic closure. Is the morphism $X \to Y$ uniquely determined by this map?
I'm asking because in the language of varieties, the map $X(k^{\text{alg}}) \to Y(k^{\text{alg}})$ is all the data you need to define a morphism (since then the $k^{\text{alg}}$-points are the whole underlying topological space), so I expect the same to be true in my case.
I have problems like this more often, by which I mean I do not really know what holds in the ordinary language of varieties versus in the language of varieties as schemes, or how to express something in one language versus the other language. This can be frustrating when the texts I am reading sometimes use the language of varieties and other times the language of schemes.
So, as a second question, I'd like to ask if there are textbooks (or any other kind of reference) that compare the two languages and make precise statements about how they relate. For example it'd be nice if you could prove that the category of varieties over $k$ (with $k$ not necessarily algebraically closed) is equivalent to a suitable full subcategory of the category of schemes over $k$.
This question turned out a bit long so thanks for reading, any help is appreciated.
