Consider $R$ a commutative ring and $M$ a $R$-module.
I know that the diagonal homomorphism is defined as follows:
$\Delta_M : M \longrightarrow M \oplus M : m \longrightarrow (m,m)$
I also know that the codiagonal homomorphism is defined as:
$\sigma_M : M \oplus M \longrightarrow M : (m,m') \longrightarrow m+m'$
Now, I want to see which are the diagonal and codiagonal morphisms in $\mathcal{C}$ an additive category. I want to justify why the objects and morphisms used in this definition exists. Any help?
The key thing to note is that additive categories have finite biproducts (simultaneous products and coproducts).
In a general abelian category then, taking the pair of maps $\newcommand\id{\mathrm{id}}\id_M,\id_M$ forms a pair of maps from $M$ to the pair of objects $(M,M)$, and hence induces a map $\Delta_M : M\to M\oplus M$, in the case of modules over a ring, you can check that this map obtained from the universal property agrees with the map you gave in your question.
Similarly this same pair of maps is also a pair of maps from the pair of objects $(M,M)$ to $M$, and hence induces a map $\sigma_M:M\oplus M\to M$, and once again you'll see that the universal map agrees with the map that you've given in the case of modules over a ring.