Matrix Multiplication by Peter Freyd I am trying to read “Abelian Categories” of Peter J. Freyd. On page 49, I came across this remark: “The usual rules of matrix multiplication can now be proven”. What does he mean? For instance, I expect to verify
$\left ( \begin{matrix} w&x\\y&z \end{matrix} \right ) 
\left (\begin{matrix} a\\b \end{matrix} \right ) =
\left ( \begin{matrix} wa+xb\\ya+zb \end{matrix} \right )$
However, if $a$, $b$, $w$, $x$, $y$, and $z$ are all maps $A \to A$, then
$\left ( \begin{matrix} w&x\\y&z \end{matrix} \right )$
is a map $A \oplus A \to A \oplus A$ and
$\left (\begin{matrix} a\\b \end{matrix} \right )$
is a map $ A \oplus A \to A$, so this composition is not defined.
The composition
$\left ( \begin{matrix} w&x\\y&z \end{matrix} \right ) 
\left (\begin{matrix} a&b \end{matrix} \right ) =
\left ( \begin{matrix} wa+yb\\xa+zb \end{matrix} \right )$
does exist, because $\left (\begin{matrix} a&b \end{matrix} \right ) $ is a map $A \to A \oplus A$. But is this a usual rule of matrix multiplication?
So, my question is, which rules does he mean and how can they be verified, what is the technique? And please, if possible, in the notation of Freyd.
 A: I managed to verify several equations of the matrix multiplication meant by Freyd. This deals with morphisms in an abelian category. In the text before Freyd’s remark, Freyd introduced addition in abelian categories, so now we can add maps
$A \overset {a} \longrightarrow X$ and
$A \overset {b} \longrightarrow X$ to a map
$A \overset {a+b} \longrightarrow X$
in an abelian category. Below, I verified one of the equations. We have
For maps
$A \overset {x} \longrightarrow X$,
$B \overset {y} \longrightarrow X$,
$X \overset {h} \longrightarrow H$,
$$h \circ
\left ( \begin{matrix} x\\y \end{matrix} \right ) =
\left (\begin{matrix} hx\\hy \end{matrix} \right )
:A \oplus B \longrightarrow H
$$
For maps
$H \overset {h} \longrightarrow X$,
$X \overset {a} \longrightarrow A$,
$X \overset {b} \longrightarrow B$,
$$
\left ( \begin{matrix} a&b \end{matrix} \right ) 
\circ h =
\left (\begin{matrix} ah&bh \end{matrix} \right ) 
:H \longrightarrow A \oplus B
$$
For maps
$K \overset {a} \longrightarrow A$,
$K \overset {b} \longrightarrow B$,
$A \overset {x} \longrightarrow Q$,
$B \overset {y} \longrightarrow Q$,
$$
\left ( \begin{matrix} x\\y \end{matrix} \right ) 
\circ
\left (\begin{matrix} a&b \end{matrix} \right ) =
xa+yb
:K \longrightarrow Q
$$
For maps
$X \overset {x} \longrightarrow Q$,
$Y \overset {y} \longrightarrow Q$,
$Q \overset {a} \longrightarrow A$,
$Q \overset {b} \longrightarrow B$,
$$
\left (\begin{matrix} a&b \end{matrix} \right )
\circ
\left ( \begin{matrix} x\\y \end{matrix} \right ) =
\left ( \begin{matrix} ax&bx\\ay&by \end{matrix} \right ) 
:X \oplus Y \longrightarrow A \oplus B
$$
For maps
$K \overset {a} \longrightarrow A$,
$K \overset {b} \longrightarrow B$,
$A \overset {w} \longrightarrow X$,
$A\overset {x} \longrightarrow Y$,
$B \overset {y} \longrightarrow X$,
$B \overset {z} \longrightarrow Y$,
$$
\left ( \begin{matrix} w&x\\y&z \end{matrix} \right ) 
\circ
\left (\begin{matrix} a&b \end{matrix} \right ) =
\left ( \begin{matrix} wa+yb&xa+zb \end{matrix} \right )
:K \longrightarrow X \oplus Y
$$
For maps
$A \overset {w} \longrightarrow X$,
$A\overset {x} \longrightarrow Y$,
$B \overset {y} \longrightarrow X$,
$B \overset {z} \longrightarrow Y$,
$X \overset {c} \longrightarrow Q$,
$Y \overset {d} \longrightarrow Q$,
$$
\left ( \begin{matrix} c\\d \end{matrix} \right ) 
\circ
\left ( \begin{matrix} w&x\\y&z \end{matrix} \right ) = 
\left ( \begin{matrix} cx+dx\\cy+dz \end{matrix} \right )
:A \oplus B \longrightarrow Q
$$
For maps
$K \overset {a} \longrightarrow A$,
$K \overset {b} \longrightarrow B$,
$A \overset {w} \longrightarrow X$,
$A\overset {x} \longrightarrow Y$,
$B \overset {y} \longrightarrow X$,
$B \overset {z} \longrightarrow Y$,
$X \overset {c} \longrightarrow Q$,
$Y \overset {d} \longrightarrow Q$,
$$
\left ( \begin{matrix} c\\d \end{matrix} \right ) 
\circ
\left ( \begin{matrix} w&x\\y&z \end{matrix} \right )
\circ
\left (\begin{matrix} a&b \end{matrix} \right ) =
cwa+dxa+cyb+dzb
:K \longrightarrow Q
$$
For maps
$G \overset {a} \longrightarrow A$,
$G\overset {b} \longrightarrow B$,
$H \overset {c} \longrightarrow A$,
$H \overset {d} \longrightarrow B$,
$A \overset {w} \longrightarrow X$,
$A\overset {x} \longrightarrow Y$,
$B \overset {y} \longrightarrow X$,
$B \overset {z} \longrightarrow Y$,
$$
\left ( \begin{matrix} w&x\\y&z \end{matrix} \right ) 
\circ
\left ( \begin{matrix} a&b\\c&d \end{matrix} \right ) =
\left ( \begin{matrix} wa+yb&xa+zb\\wc+yd&xc+zd \end{matrix} \right ) 
:G \oplus H \longrightarrow X \oplus Y
$$
For maps
$X \overset {x} \longrightarrow H$,
$X\overset {y} \longrightarrow H$,
$H \overset {h} \longrightarrow Q$,
$Q \overset {a} \longrightarrow A$,
$Q \overset {b} \longrightarrow B$,
$$
\left (\begin{matrix} a&b \end{matrix} \right ) 
\circ h \circ
\left ( \begin{matrix} x\\y \end{matrix} \right ) =
\left ( \begin{matrix} ahx&bhx\\ahy&bhy \end{matrix} \right )
:X \oplus Y \longrightarrow A \oplus B
$$
As an example, we verify for maps
$K \overset {a} \longrightarrow A$,
$K \overset {b} \longrightarrow B$,
$A \overset {x} \longrightarrow Q$,
$B \overset {y} \longrightarrow Q$,
that
$$
\left ( \begin{matrix} x\\y \end{matrix} \right ) 
\circ
\left (\begin{matrix} a&b \end{matrix} \right ) =
xa+yb
:K \longrightarrow Q
$$
as follows:
There is a unique
$q = \left (\begin{matrix} a&b \end{matrix} \right ) :K \longrightarrow A \oplus B$
such that
$p_1 \circ q = a$ and $p_2 \circ q = b$
There is a unique
$t = \left (\begin{matrix} x\\y \end{matrix} \right ) :A \oplus B \longrightarrow  Q$
such that
$t \circ u_1 = x$ and $t \circ u_2 = y$
Then we have
$$t \circ q = $$
$$ 
= \left (\begin{matrix} x\\y \end{matrix} \right )
\circ
\left (\begin{matrix} a&b \end{matrix} \right ) = 
$$
$$ 
= \left (\begin{matrix} x\\y \end{matrix} \right )
\circ 1_{A \oplus B} \circ
\left (\begin{matrix} a&b \end{matrix} \right ) = 
$$
$$ 
= \left (\begin{matrix} x\\y \end{matrix} \right )
\circ \left (  u_1p_1+u_2p_2 \right ) \circ
\left (\begin{matrix} a&b \end{matrix} \right ) =
$$
$$
= \left ( xp_1+yp_2  \right )
\circ
\left (\begin{matrix} a&b \end{matrix} \right ) = 
$$
$$
=xa+yb
$$
