I can't solve Exercise 3.22 from W. Lawvere - Sets For Mathematics book.
Exercise 3.22. Show that for any $A$, $1 \times A \simeq A$.
The exercise has the following hint.
Hint: To show one of the equations requires using the uniqueness clause in the definition of product.
One page earlier the author gives the following definition
Definition 3.20. In any category, a product of two objects $A$ and $B$ is a pair of giving mappings $$ A \xleftarrow{\pi_A} P \xrightarrow{\pi_B} B$$ such that $$\forall X,\ f\colon X \to A,\ g\colon X \to B\ \ \exists! h\colon X \to P\ [f = \pi_Ah \text{ and } g = \pi_Bh]$$ as in the following commutative diagram.
I'm trying to replace $X$ with $A$, $B$ with $1$ and $f$ with $1_A$ in the diagram above. It gives me $\pi_Ah = 1_A$. Now I'm stuck at showing $h\pi_A =1_{A \times 1}$.
$h\pi_A : 1\times A\to 1\times A$ is entirely determined by the composites with $\pi_1$ and $\pi_A$, by the uniqueness part of the definition of product.
Now $1$ is terminal, so $\pi_1\circ $whatever can only be one thing.
What is $\pi_A\circ h \circ \pi_A$ ?
Do you know another interesting map $1\times A\to 1\times A$ that has the same projections ?