Wedge product of a direct sum and the Yoneda Lemma In a comment to https://math.stackexchange.com/a/344851/58601, Martin Brandenburg suggests that one may prove the existence of the canonical isomorphism $\wedge^n(W_1 \oplus W_2) \to \bigoplus_{p+q=n} \wedge^p(W_1) \otimes \wedge^q(W_2)$ with the Yoneda Lemma. Is there a reference for this technique?
I'm interested in a more conceptual proof of the fact that the map is an isomorphism than the "pick a basis" proof.
 A: Concerning a conceptual proof, the trick is to put enough structure on both sides: They are not only ${\mathbb k}$-modules, but the homogeneous components of the graded-commutative ${\mathbb Z}$-graded ${\mathbb k}$-algebras ${\bigwedge}^{\ast} (W_1\oplus W_2)$ and $({\bigwedge}^{\ast} W_1)\otimes_{\mathbb k} ({\bigwedge}^{\ast} W_2)$, respectively, where for two graded-commutative ${\mathbb Z}$-graded ${\mathbb k}$-algebras $A$ and $B$ the $\otimes_{\mathbb k}$-product $A\otimes_{\mathbb k} B$ is defined via $(A\otimes_{\mathbb k} B)^n := \bigoplus\limits_{p+q=n} A^p\otimes_{\mathbb k} B^q$ and multiplication $(a\otimes b)(a^{\prime}\otimes b^{\prime}) := (-1)^{|b|\cdot |a^{\prime}|} (aa^{\prime})\otimes (bb^{\prime})$ and constitutes the coproduct of $A$ and $B$ in the category ${\mathbb Z}\text{-grcomm-alg}_{\mathbb k}$ of graded-commutative ${\mathbb Z}$-graded ${\mathbb k}$-algebras. The claim then follows from the observation that the assignment $V\mapsto{\bigwedge}^{\ast} V$ defines a left-adjoint ${\mathbb k}\text{-Mod}\to{\mathbb Z}\text{-grcomm-alg}_{\mathbb k}$ to the forgetful functor ${\mathbb Z}\text{-grcomm-alg}_{\mathbb k}\to{\mathbb k}\text{-Mod}$ sending a ${\mathbb Z}$-graded, graded-commutative ${\mathbb k}$-algebra $A$ to its degree $1$ component $A^1$. In particular, it preserves arbitrary colimits, and in particular coproducts.
The Yoneda-Lemma is implicit here in the proof that left adjoints preserve colimits.
