Linear categories in Lawvere's Conceptual Mathematics

The product $X\times Y$ implicitly comes with two fixed projections $\pi_X,\pi_Y$, and these play an implicit role in the definitions.
Dually, the coproduct $X+Y$ comes equipped with inclusions $\iota_X,\iota_Y$.

First, observe that (writing composition to the right), $$\underset{X+Y\,\to\, X\times Y\,\to\, X}{\pmatrix{a&b\\c&d}\pi_X}\ =\ \pmatrix{a\\c}$$ and similarly, composing it by $\pi_Y$ yields the second column.
Consequently, as $1_{X\times Y}=\alpha\,\pmatrix{1&0\\0&1}$, we have $$ \pi_X\ =\ \alpha\,\pmatrix{1&0\\0&1}\pi_X\ =\underset{X\times Y\to X+Y\to X}{\qquad \alpha\,\pmatrix{1\\0}}\,. $$ Dual statements hold for rows and the $\iota$'s.

From here, we can evaluate the 3 required elements like: $$\iota_Y\cdot\,\pmatrix{1&f\\0&1}\alpha\pmatrix{1&g\\0&1}\,\cdot\pi_X\ =\ \pmatrix{0&1}\,\alpha\,\pmatrix{1\\0}\ =\ \pmatrix{0&1}\pi_X\ =\ 0\,.$$