Solution I.5.11 incorrect?

[wanderinglethe]

Thank you for publishing these solutions, it has helped me understanding the material better.

I have a question about I.5.11, in the first part you do not use the definition of $\sim$ and do not show $\pi^A_{\sim_A}\pi_A$ is an morphism that satisfies the quotient condition, is this implicit?

Then in the second part I do not see how $1_A$ would satisfy the quotient condition $A/{\sim_A}$ since that only seems to work if $\sim_A = I_A$ (the identity relation). What am I missing here?

Thank you