I have trouble understanding the differences between the following two equivalent definitions of algebraic spaces. Let $k$ be a field.
The first one is
An algebraic space is an '{e}tale sheaf $(Sch/S)^{opp}_{et}\rightarrow Sets$ such that some axioms are satisfied.
The second one is
An algebraic space is an '{e}tale sheaf $k{\textrm{-alg}}\rightarrow Sets$ such that some axioms are satisfied.
How can we transform the first definition to the second one? Or how can we transform the second definition to the first one? More precisely, why $(Sch/S)^{opp}_{et}$ becomes $k{\textrm{-alg}}$?
Here is the complete definition of the first one:
And this is the second one ($\textrm{Aff}_{k}$ is the category of $k$-algebras):
