Affine-target adjunction for varieties ($\mathrm{Hom}(X,Y)\cong {\mathrm{Hom}}_{k\text{-alg}}(k[Y],\Gamma (X,{\mathcal{O}}_X))$)

For a classical algebraic variety $X$ and affine variety $Y$, a homomorphism on the indicated rings gives a morphism by evaluating the images of affine coordinate generators at each point of $X$. The relations defining $Y$ remain zero. Local fractions pull back to local regular fractions, proving regularity; the coordinate ring generators also prove uniqueness.