Characteristic property of the direct sum module/Statement

Statement

$\begin{xy} \xymatrix{ \bigoplus_{\alpha\in I}M_\alpha \ar[ddrr]^\varphi \ar@{<-_{)} }[dd] & & \\ & & \\ M_b \ar[rr] & & M \save (15,13)+"3,1"*+{\ldots}="udots"; (8.125,6.5)+"3,1"*+{M_c}="x1"; (-8.125,-6.5)+"3,1"*+{M_a}="x3"; (-15,-13)+"3,1"*+{\ldots}="ldots"; \ar@{<-_{)} } "x1"; "1,1"; \ar@{<-_{)} }_(0.65){i_c,\ i_b,\ i_a} "x3"; "1,1"; \ar@{<-} "x1"; "3,3"; \ar@{<-}^{\varphi_a,\ \varphi_b,\ \varphi_c} "x3"; "3,3"; \restore } \end{xy}$

TODO: Caption

Let $(R,+,*,0)$ be a ring (with or without unity) and let $(M_\alpha)_{\alpha\in I}$ be an arbitrary indexed family of $R$-modules and $\bigoplus_{\alpha\in I}M_\alpha$ their direct sum (external or internal). Let $M$ be another $R$-module. Then:

• For any family of module homomorphisms, $(\varphi:M_\alpha\rightarrow M)_{\alpha\in I}$
  • There exists a unique module homomorphism, $\varphi:\bigoplus_{\alpha\in I}M_\alpha\rightarrow M$, such that
    • $\forall\alpha\in I[\varphi\circ i_\alpha=\varphi_\alpha]$

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TODO: Mention commutative diagram and such

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