Associativity of the additive law in a category with finite biproducts

Question

It is well known that if $\mathcal{C}$ is a category with finite biproducts, then we can define a binary operation "$+$" on every set of morphisms $Hom_{\mathcal{C}}(X,Y)$ using the diagonal $\Delta_{X}:X\rightarrow X\oplus X$ and codiagonal $\nabla_{Y}:Y\oplus Y\rightarrow Y$ morphisms; and it turns out that with this operation the sets $Hom_{\mathcal{C}}(X,Y)$ are commutative monoids, where the identity of the operation is the zero morphism $0:X\rightarrow Y$. I've been able to show that the operation is commutative and that indeed the zero morphism is the identity for the operation. However, I'm having a lot of troubles trying to show the associativity of the operation. Instead of an answer, I'd like a hint to solve this by myself.

Answer 1

Hint: Show that both $f+(g+h)$ and $(f+g)+h$ equal $$X \xrightarrow{\Delta} X \oplus X \oplus X \xrightarrow{f \oplus g \oplus h} Y \oplus Y \oplus Y \xrightarrow{\nabla} Y$$ via some commutative diagramm.