Characteristic property of the direct sum module/Statement/Picture

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

TODO: Caption