This is more of a moral (i.e. category theoretical) question.
In the category of $R$-modules for a ring $R$, the product is the direct product $M=\prod_{i\in I}M_i$ with canonical projections $\pi_i\colon M\twoheadrightarrow M_i$ for all $i\in I$. Similarly, the coproduct is the direct sum $M=\bigoplus_{i\in I}M_i$ with canonical inclusions $\iota_i\colon M_i\hookrightarrow M$ for all $i\in I$.
But why do direct sum admit projections, i.e. $\bigoplus_{i\in I}M_i\twoheadrightarrow M_j$ for each $j\in I$? Similarly, why do direct products admit inclusions, i.e. $M_j\hookrightarrow\prod_{i\in I}M_i$ for each $j\in I$?
By the universal property of the direct sum as a coproduct, there exists a unique morphism $\pi_j : \bigoplus_i M_i \to M_j$ such that $\pi_j \circ \iota_i = 0$ if $i \ne j$ and $\pi_j \circ \iota_i = \operatorname{id}_{M_j}$ is $i = j$.
Similarly, by the universal property of the product, there exists a unique morphism $\iota_j : M_j \to \prod_i M_i$ such that $\pi_i \circ \iota_j = 0$ if $i \ne j$ and $\pi_i \circ \iota_j = \operatorname{id}_{M_j}$ if $i = j$.
(Note that either argument can easily be generalized in any complete/cocomplete category such that $\operatorname{Hom}(X, Y)$ is nonempty for each pair of objects $X$ and $Y$. And if there is a canonical choice of an element of each hom-set -- as in the case of the zero morphism in any preadditive category, or any category with a zero object -- then this construction becomes a canonical one.)