TITLE: A proof using Yoneda lemma
QUESTION [18 upvotes]: Martin Brandenburg pointed out elsewhere in the comments that he could give a one line proof, using the Yoneda lemma, of
$$\frac{\mathbf{C}[x_1,\ldots,x_{n+m}]}{I(X)^e+I(Y)^e} \cong \frac{\mathbf{C}[x_1,\ldots,x_n]}{I(X)} \otimes_\mathbf{C} \frac{\mathbf{C}[x_{n+1},\ldots,x_{n+m}]}{I(Y)}$$ (for $X,Y$ affine algebraic varieties), but apparently the proof was too long for his margin. How can this be done?
Fundamental question aside: why is there no abstract nonsense tag?

REPLY [17 votes]: $$\hom(R/I \otimes S,-) = \hom(R/I,-) \times \hom(S,-) = \{f \in \hom(R,-):f(I)=0\} \times \hom(S,-)$$
$$ = \{h \in \hom(R \otimes S,-) : h(I \otimes 1)=0\}=\hom((R \otimes S)/I^e,-).$$
Hence, $R/I \otimes S = (R \otimes S)/I^e$. Then also $R/I \otimes S/J = (R \otimes S/J)/I^e = (R \otimes S)/(J^e,I^e)$.