Question about products and coproducts I am confused as to why this is true, or how to obtain it: $Hom(\bigoplus_{j} H_k(X_j), \mathbb{R}) \cong \prod_{j} Hom(H_k(X_j), \mathbb{R})$. How would one prove this relation, specifically without it being very involved if possible? This question arises is arising from Proposition 18.5 in John Lee's Smooth Manifolds, if context helps. Any help would be appreciated. If possible, an explanation at a more beginner friendly level would be appreciated (I am an undergrad and am doing this to finish up a project for a program from summer). I'm still struggling with the definitions, as I cannot seem to fine ones that are understandable to my level.
 A: Appendix B, Proposition B.55 in your book (John Lee's Introduction to Smooth Manifolds) gives the characteristic property of the direct sum, which is that direct sum has the property of being a coproduct.
Applied to your case, a coproduct $\bigoplus_j H_k(X_j)$ by definition comes equipped with a family of maps $\iota_j\colon H_k(X_j)\to\bigoplus_j H_k(X_j)$ so that for any other family of maps $f_j\colon H_k(X_j)\to R$ there exists a unique map $f\colon \bigoplus_jH_k(X_j)\to R$ satisfying $f\circ\iota_j=f_j$ for each $j$.
Then Proposition B.57 proves the relation you are asking about is an immediate consequence of the characteristic property of direct sums.
In particular, if for each $R$ and each $f\in Hom(\bigoplus_jH_k(X_j),\mathbb R)$ we denote by $\gamma_R(f)$ the family $(f\circ\iota_j)\in\prod_jHom(H_k(X_j),R)$, then the uniqueness of $f$ in the definition of corpoduct asserts that $\gamma_R\colon Hom(\bigoplus_jH_k(X_j),\mathbb R)\to\prod_jHom(H_k(X_j),\mathbb R)$ is one-to-one, while the existence of $f$ in the definition asserts that $\gamma_R\colon Hom(\bigoplus_jH_k(X_j),\mathbb R)\to\prod_jHom(H_k(X_j),\mathbb R)$ is onto. Combining the two claims, $\gamma_R$ is by definition a bijrection.
