# embedding a k-linear category in an additive category

I am reading an article and it is written there: If A is a k-linear category (possibly without direct sums) we can embed it in the additive category A × N, where a morphism (x,m) → (y, n) is an n × m matrix with entries in A(x, y) = HomA(x, y). Of course if A is additive then A ≈ A × N. Could anyone explain it? Why is it additive?

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Well, you haven't got the objects quite right. A better description of the objects of the free additive category $\oplus A$ over $A$ is: $n$-tuples of objects $x = (x_{1}, \ldots, x_{n})$ of $A$ where $n \in \mathbb{N}$. Now a morphism in $\text{Hom}\,(x,y)$ is a matrix $(f_{ij})$, where $f_{ij} \in \text{Hom}_A\,(x_{j},y_{i})$ and composition is given by the familiar formula for matrix multiplication.
1. every additive functor $F: A \to B$ to an additive category $B$ extends uniquely to an additive functor $\oplus F : \oplus A \to B$.
2. If $A$ is already additive then $A$ and $\oplus A$ are equivalent.
3. What is the free $k$-linear category over $A$?