Let $\mathcal{A}$ be an additive category and $\text{Arr}(\mathcal{A})$ be its arrow category i.e. the objects are the morphisms in $\mathcal{A}$ and the morphisms from $u:A\to B$ to $v:C\to D$ are the pairs $(f,g)$, where $f:A\to C$, $g:B\to D$ are morphisms in $\mathcal{A}$ and $vf=gu$ (so they form commuting squares).
I am trying to make sense of the additive structure on $\text{Arr}(\mathcal{A})$ inherited from $\mathcal{A}$, in particular the biproducts of objects $u$ and $v$. I assume this must be the 'direct sum' of morphisms in $\mathcal{A}$ i.e. the morphism $u\oplus v:A\oplus C\to B\oplus D$, but I cannot make sense of this in a strictly categorical context.
We have morphisms $A\oplus C\to A\to B\to B\oplus D$ and $A\oplus C\to C\to D\to B\oplus D$ via $u$ and $v$ and using the inclusions/projections and I assume these are meant to be equal but how do I show this?