On the page : $ 58 $ of the following pdf: http://rgug.ch/medias/math/geometrie_algebrique.pdf , we find the proof of the theorem which says that there exists a fully faithful functor $ t $ from the category of varieties over $ K $ to the category of $ \mathrm{Spec} K $ - schemes : $ t: \mathrm{V}^K \to \mathcal{S} \mathrm{ch} (K) $.
The author of this proof tells us how to built this functor $t$, but the most important thing which is to show that $ t $ is fully faithful, which means that : $ \Phi: \mathrm {Hom}_{\mathbb{V}^{K}} (V, W) \to \mathrm{Hom}_{\mathcal{S} \mathrm{ch}} (t (V), t ( W)) $ is a natural bijection is not made in this pdf, look at the page: $ 60 $ to understand.
Hence, my question is :
In which book can i find a rest of this proof which says that: $ \Phi: \mathrm{Hom}_{\mathbb{V}^{K}} (V, W ) \to \mathrm{Hom}_{\mathcal{S} \mathrm{ch}} (t (V), t (W)) $ is a natural bijection ( e.g : Injectivity + Surjectivity) ?
Thanks in advance for your help.