I am not sure the example you choose is the most striking application of category theory in algebraic geometry
However, there is a similar question where indeed category theory makes the heart of the phenomenon way clearer.
Let $ f : X \longrightarrow Y$ be a proper morphism of schemes. You get a functor $Rf_* : D(X) \longrightarrow D(Y)$ known as the derived push-forward (here $D(X)$ is the (unbounded) derived category of quasi-coherent sheaves on $X$). It is very natural to ask when this functor has a right adjoint.
If you make some serious hypothesis on the singularities of $X$ and $Y$, Grothendieck stated a general existence Theorem for this right-adjoint (this is Grothendieck-Serre duality). The proof of this duality statement as envisioned by the Master is very complicated and proceeds by identifying precisely who is the right adjoint. A complete proof has been written by Hartshorne in Residues and Duality. I personnally find it very hard to understand.
On the other hand, in the mid 90ies, Neemann found a beautiful and very simple categorical proof of the existence of such a right adjoint using Brown representability Theorem. This is beautifully written in Grothendieck Duality via Bousfield's localization and Brown representability.
Note however that there is minor drawback in Neeman approach : even if $X$ and $Y$ are smooth projective, you don't get a formula for the right adjoint. However, knowing that it exists allows you to find an explicit description of it quite quickly in the Gorenstein case.