Why do schemes give sheaves on Etale site?

I am learning about sites and sheaves on them. The first significant example is the big Etale site on all $k$-schemes.

Let $Y$ be a scheme and $h_Y=Hom(-,Y)$ be the corresponding contravariant functor to $Sets$.

How do I show that it is a sheaf on the Etale site?

The question is essentially about given Etale covers $X_i,X_j$ of $X$ and fibre product $X_{ij}$ and morphisms from $X_i$ and $X_j$ to Y which agree on $X_{ij}$ how do I extend it to a morphism from $X \to Y $ ?

I believe if I take everything involved to be affine then it reduces to some commutative algebra question. I am unable to solve that corresponding question.

Solution 1:

Question: "I believe if I take everything involved to be affine then it reduces to some commutative algebra question. I am unable to solve that corresponding question."

Answer: You find a self contained exaplanation in Milne's "Etale cohomology" (it requires commutative algebra at the level of Atiyah-Macdonald.