Let $C$ be a commutative ring and $E_1,E_2,F_1,F_2$ $C$-modules. Suppose $E_1$ and $F_1$ are projective and finitely generated. I want to show that the canonical mapping
$$\lambda:\text{Hom}(E_1,F_1)\otimes\text{Hom}(E_2,F_2)\rightarrow\text{Hom}(E_1\otimes E_2,F_1\otimes F_2)$$
is an isomorphism. Is there a short categorical proof of this? The proof I have has 4 lemmas and is almost two pages; it implicitly uses functors and natural transformation. It treats $\lambda$ as a natural transformation between the functor $\text{Hom}(-,F_1)\otimes\text{Hom}(E_2,F_2)$ and $\text{Hom}(-\otimes E_2,F_1\otimes F_2)$ and also as natural transformation between $\text{Hom}(C,-)\otimes\text{Hom}(E_2,F_2)$ and $\text{Hom}(C\otimes E_2,-\otimes F_2)$.
Any way to clean this proof up?
Best Answer
First prove it for $E_1 = F_1 = C$ where it's very easy. Then prove that if it's true for two values of $E_1$ then it's true for their direct sum, and similarly for $F_1$; this proves it for $E_1, F_1$ finitely generated free. Then prove that if it's true for some value of $E_1$ then it's true for any retract of it, and similarly for $F_1$; this proves it for $E_1, F_1$ finitely generated projective.
The general pattern here is that "retracts are absolute colimits," which means they are preserved by any functors whatsoever. See the section titled "The facts of life about idempotents and retracts" in my blog post Tiny objects for details.