The problem is to show that in the category of real vector spaces the direct product of countably infinitely many $\mathbb{R}$ is isomorphic to $\mathbb{R}[[t]]:= \sum\limits_{j=1}^\infty a_j t^j, \,\,a_j\in \mathbb{R}$, and the direct sum to polynomials $\mathbb{R}[t]$.
It's not that I don't understand the problem, or that it's not obvious to me; what I am is very uncomfortable with category theory, and pessimistic regarding my attempt to do it in a category-theoretic manner. So here goes: defining the projection maps thus: $\pi_i: \mathbb{R}[[t]] \longrightarrow \mathbb{R}, \quad \sum\limits_{j=1}^\infty a_j t^j \longmapsto a_i,$ $p_i: \prod\limits_{j=1}^\infty \mathbb{R} \longrightarrow \mathbb{R}, \quad (a_1, \cdots,a_j,\cdots)\longmapsto a_i,$ by the universal property of products, $\mathbb{R}[[t]] \cong \prod\limits_{j=1}^\infty \mathbb{R}$. The case of $\mathbb{R}[t] \cong \bigoplus\limits_{j=1}^\infty$ follows from this when all but finitely many of the $a_i = 0$.
Is this wrong? Should I not have done this?