Suppose I have a commutative diagram $D$ in the category of abelian groups. In $D$ there appear abelian groups $A$, $B$, $A\oplus B$ and some other abelian groups. Also, the natural inclusions $i:A\to A\oplus B$ and $j:B\to A\oplus B$ are also present in $D$. Suppose there is an abelain group $X$ in $D$ on which we have homolomorphisms $f:A\to X$ and $g:B\to X$ appearing in the diagram but there is, at present, no arrow from $A\oplus B$ to $X$ in $D$.
By the co product property of $A\oplus B$, there is a unique arrow from $h:A\oplus B\to X$ such that $h\circ i=f$ and $h\circ j=g$.
Question 1. Can adding $h$ to $D$ disturb the commutativity of $D$?
I think the the answer to the above question in NO.
Here is my reasoning. First we note that if $I$ is an initial object in a category $\mathcal C$, and $D$ is a commutative diagram in $\mathcal C$ which features $I$, then one can freely put an arrow starting at $I$ to any object $X$ in $D$ (if such an arrow is not already there) without disrupting the commutativity.
Now the direct sum of two abelian groups can be thought of as an initial object in the category of "cones" of $A$ and $B$. In other words, consider the category where the objects are triples $(X, f:A\to X, g:B\to X)$ and a morphism between $(X, f:A\to X, g:B\to X)$ and $(Y, h:A\to Y, k:B\to Y)$ is a homomorphism $F:X\to Y$ making the corresponding diagram commute.
Then by the previous remark about initial objects, we can answer the above question with a NO.
Question 2. Can we more generally say that if there is a universal object in a commutative diagram, then we can draw any missing arrows which can be drawn by using the universal property without disturbing the commutativity of the diagram.
Can somebody answer the above question?