Let $\mathcal{C}$ be an additive category. I define diagonal and codiagonal morphisms as follows:
Diagonal:
Let $A \in Ob(\mathcal{C})$. Then the diagonal $\Delta_A: A \longrightarrow A \oplus A$ is the unique morphism with $p \circ \Delta_A = id_A$ and $q \circ \Delta_A = id_A$ where p,q are projections from $A \oplus A$ to $A$.
Codiagonal:
Let $B \in Ob(\mathcal{C})$. Then the codiagonal $\nabla_B: B \oplus B \longrightarrow B$ is the unique morphism with $\nabla_B \circ i= id_B$ and $\nabla_B \circ j = id_B$ where i,j are injections from $B$ to $B \oplus B$.
I would like to know how can I justify the existence of these objects and morphisms. Any help?
The objects I must verify their existence are: $A$, $A \oplus A$.
The morphisms I must verify: $id_A$, projections and injections.
Best Answer
All of this follows directly from the definition of an additive category, together with the definition of all of the terms in that definition. The most complicated part of that definition is understanding in detail what it means for a category to admit biproducts; for that see this blog post.