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

The assertion is not true, as monomorphisms and epimorphisms in MSet (as defined) are precisely the injective and surjective maps (respectively) in Set that respect the equivalence relations.

PDf

x = p((p − id_M)x) ∈ M is an incorrect assertion.