Distributive functor over the biproduct I  have just been introduced to  abelian categories, so my doubt should  be quite trivial; I hope not too much though.
Given two biproducts in an abelian category  $\mathcal  A$, say $A_1\oplus A_2$ and $A_3\oplus A_4$, the morphisms $f:A_1\oplus A_2\to A_3\oplus A_4$ can be represented as  matrices $2\times  2$ in the known way. Tensoring for some  object $A_0$ yields a  morphism  $$(A_1\otimes A_0)\oplus (A_2\otimes A_0)\to (A_3\otimes A_0)\oplus (A_4\otimes A_0),$$ and this is true because the tensor distributes over the biproduct. But very likely is true also that, if $$\begin{bmatrix}
    a       & b \\
    c       & d \\
    \end{bmatrix}$$
is the matrix of some  morphism $f$ as above, the matrix of  $f\otimes  1_{A_0}$ is
$$\begin{bmatrix}
    a\otimes  1_{A_0}       & b\otimes  1_{A_0} \\
    c\otimes  1_{A_0}       & d\otimes  1_{A_0} \\
    \end{bmatrix}$$
The situation, as I  pictured,  is that if one has an abelian functor $F:\mathcal  A\to \mathcal A$ can always calculate $Ff$ summing
$$F\begin{bmatrix}
    a       & 0 \\
    0       & 0 \\
    \end{bmatrix}+F\begin{bmatrix}
    0       & b \\
    0       & 0 \\
    \end{bmatrix}+\dots$$
However if $F$ doesn't distribute  over the biproduct  it doesn't  make  sense to  say that even $$F\begin{bmatrix}
    a       & 0 \\
    0       & 0 \\
    \end{bmatrix}=\begin{bmatrix}
    Fa       & 0 \\
    0       & 0 \\
    \end{bmatrix}$$
or that
$$F\begin{bmatrix}
    a       & b \\
    c       & d \\
    \end{bmatrix}=\begin{bmatrix}
    Fa       & Fb \\
    Fc       & Fd \\
    \end{bmatrix}$$
Being not practical of abelian categories,  I couldn't think of an example  of an abelian functor that doesn't distribute over the biproducts, but I believe there are. My very  question is: knowing only that an abelian functor distributes over the (object) components of the biproduct, can one prove that the functor distributes also over the morphisms, like the tensor does? (That under this assumption for $F$ now makes  sense as a question, at least). My guess would be to compute the matrix of $Ff$, that is unique, and show that the components are $Fa,Fb,Fc,Fd$;  but it seems obvious, because, for example, the first component is obtained as
$$F(\pi_{A_3})F\begin{bmatrix}
    a       & b \\
    c       & d \\
    \end{bmatrix}F(\iota_{A_1})=F(\pi_{A_3}\begin{bmatrix}
    a       & b \\
    c       & d \\
    \end{bmatrix}\iota_{A_1})=Fa$$
where $\iota_{A_1}:A_1\to A_1\oplus A_2$ and $\pi_{A_3}:A_3\oplus A_4\to A_3$ are the canonical morphisms of the biproducts. Of course we need at least to assume that such canonical morphisms are preserved by $F$, in addition to distributing over the objects of the biproduct. What  do you think about this discourse, is it a mess or does it  make sense?
 A: For an ordinary functor $F:\cal A\rightarrow\cal B$ between abelian categories $\cal A,B$ the following are equivalent

*

*$F$ preserves biproducts (or distributes over them in your terminology)

*$F$ is an $\mathsf{Ab}$-enriched functor (I guess this is what you mean with abelian functor) ie. for all objects $A,A‘$ of $\cal A$ the map $\mathcal{A}(A,A‘)\rightarrow \mathcal{B}(FA,FA‘)$ is a homomorphism of abelian groups.

This is because biproducts can be completely characterized in terms of equations (an object $C$ together with projections $p_A,p_B$ and inclusions $i_A,i_B$ satisfying $p_Bi_A=0$, $p_Ai_B=0$, $p_Ai_A=id$, $p_Bi_B=id$ and $i_Ap_A+i_Bp_B=id$) and the group structure on the hom-sets can be recovered from the biproducts via $$f+g=X \overset{\Delta}{\longrightarrow} X \oplus X \overset{f\oplus g}{\longrightarrow} Y \oplus Y \overset{\nabla}{\longrightarrow} Y$$
In particular additive functors (ordinary functors satisfying one of the two equivalent conditions) preserve the matrix representations of morphisms between biproducts and their decomposition as sum of the elementary matrices.
