A question about the definition of additive categories.

Question

What does the following line mean

The composition of morphisms is distributive over addition.

Does it mean $a+(b\circ c)=(a+b)\circ (a+c)$?

Answer 1

It means that the composition map $$ \hom(A,B) \times \hom(B,C) \to \hom(A,C),\, (f,g) \mapsto g \circ f $$ is $\mathbb Z$-bilinear for all objects $A,B,C$ in the category. (A more general concept is the one of enriched category over a monoidal category.)

So it means that $a\circ (b+c) = a \circ b + a \circ c$ and that $(a+b) \circ c = a \circ c + b \circ c$ whenever the compositions make sense.