Question about products and coproducts

Solution 1:

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.