The set of linear maps between vector spaces form a vector space, and the set of abelian group homomorphisms form an abelian group - general fact?

Question

I was recently thinking about maps between mathematical objects $f,g:X\to Y$ which have some binary operation for which we can form a new map $f+g:X\to Y$, examples including

• if $X$ and $Y$ are vector spaces, we can add two linear transformations via $(f+g)(x)=f(x)+f(y)$ using the addition on $X$. Furthermore, the set of all linear maps $X\to Y$ form a vector space.
• if $X$ and $Y$ are abelian groups, we can add two group homomorphisms $f$ and $g$ to get a new group homomorphism $(f+g)(x)=f(x)+g(x)$ in the same way, and the set of homomorphisms form an abelian group.

Is there a name for this concept? What is the minimal structure the objects $X$ and $Y$ need to have? I considered monoidal categories, but, we are combining morphisms between objects, not the objects themselves. I also considered things like the dual space, but this involves maps from the vector space to its underlying field, not between vector spaces.

Answer 1

Based on Thomas Andrews' comment: the correct idea (with minimal assumptions) seems to be preadditive category, or Ab-category:

a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear...

with examples including abelian groups and vector spaces.