That's proposition 1.2.7 in Borceux Categorical Algebra Volume 2.
What does it mean exactly?
I understand the proof that $Coker\Delta_C = p_1-p_2$ but then he says:
"which proves that the difference $p_1-p_2$ is characterized, up to isomorphism, by the limit-colimit structure of $\mathscr C$."
So what is an isomorphism of additive structures? What is the theorem really saying conceptually or morally? And how does it relate to that last sentence which I also don't understand on its own?
For anyone without the book at hand, as a reminder, $p_1 : C\oplus C \rightarrow C$ is the first projection and $p_2 : C\oplus C \rightarrow C$ is the second projection. And $\Delta_C : C\rightarrow C\oplus C$ is characterized by $p_1\circ\Delta_C = id_C = p_2\circ\Delta_C$. Also $C$ is just a fixed object in $\mathscr C$.
Thanks in advance!
First, let's mention the difference between property and structure.
A property of an object is something that it either is or isn't. For example, "being abelian" is a property of a group $G$. It either is or is not abelian, and you can tell using only the structure already present in $G$ whether or not it has that property. In particular, if $G \cong G'$ as groups, then $G$ and $G'$ are automatically isomorphic as abelian groups. Since there's only one way to be abelian!
There's also structure on an object. Which is some ~bonus data~ that an object might or might not have. For instance, a multiplication structure on a group $G$ is a structure. Note that there might be multiple different ring structures you can put on $G$ (for instance, if $G = \mathbb{Z}$ you can give $G$ the usual multiplication or the zero multiplication $\forall a, b . ab = 0$) and now a group isomorphism $G \cong G'$ is not enough to give an isomorphism as rings. You need to check by hand that your group isomorphism is compatible with the multiplication structure.
There are lots of properties in category theory. For instance "having finite products", "having coproducts", "having a terminal object", etc. These are all properties of your category since you either have finite products or you don't. For an example of a structure in category theory, take a monoidal product $\otimes$. In general you can have multiple inequivalent monodial products on the same category! So this is really a ~bonus structure~ that you have to give your category. In particular, if $(\mathcal{C}, \otimes)$ and $(\mathcal{C}', \otimes')$ are monoidal categories, and $\mathcal{C} \simeq \mathcal{C}'$, we can't immediately say that they're equivalent as monoidal categories because their monoidal structures might be different!
Now, remember that an additive category for Borceux is a category $\mathcal{A}$ with a zero object $0$, binary biproducts $\oplus$, and an abelian group structure on each homset $\mathcal{A}(X,Y)$ (satisfying certain natural axioms saying that these are all compatible). It's pretty easy to believe that $0$ and $\oplus$ are properties. You either have biproducts, or you don't. But it seems at first glance that there might be different ways to put an abelian group structure on each homset! After all, any single set $A$ has tons of different abelian group structures! And now we have multiple homsets to deal with too!
What Borceux is saying with this theorem is that, miraculously, "being an additive category" is a property! Either $\mathcal{A}$ is additive, or it isn't. So if we have two additive categories $\mathcal{A}$ and $\mathcal{A}'$, at first glance it seems like the right notion of "sameness" should be an equivalence $F : \mathcal{A} \simeq \mathcal{A}'$ plus proofs that all the abelian groups $\mathcal{A}(X,Y) \cong \mathcal{A}'(FX,FY)$ are isomorphic. The magic of this theorem is that we don't need to check the group structures! As soon as $\mathcal{A} \simeq \mathcal{A}'$ as categories, we know that the abelian groups on their homsets are automatically isomorphic!
The succinct way to say this (besides "being an additive category is a property") is to say that if $\mathcal{A}$ is additive, it's additive in a unique way. This is because you can actually phrase the group structure on $\mathcal{A}(X,Y)$ in terms of the rest of the properties of $\mathcal{A}$!
This is what Borceux is talking about with the stuff about $p_1 - p_2$. The insight is that we can recover the group structure on $\mathcal{A}(X,Y)$ from the rest of the category structure! For instance, if $f,g : X \to Y$ then one can show that $f+g$ must be the composite
$$ X \overset{\langle f, g \rangle}{\longrightarrow} Y \oplus Y \overset{\nabla}{\longrightarrow} Y $$
where $\langle f, g \rangle$ is the map guaranteed by the fact that $\oplus$ is a product, and $\nabla$ is the codiagonal guaranteed by the fact that $\oplus$ is a coproduct.
Since coproducts and composition are properties of $\mathcal{A}$, and they define the abelian group structure on each $\mathcal{A}(X,Y)$, we see that these group structures (and thus the additive-ness of $\mathcal{A}$) are also properties!
I hope this helps ^_^