In Aluffi's Algebra Chapter 0, on page 34, he proves the following claim:
Claim 5.5: Denoting by $\pi$ the 'canonical projection' defined in Example 2.6 [he means $\pi:A \rightarrow A/\sim$ for some equivalencce relation $\sim$] the pair $(\pi, A/\sim)$ is an initial object of this category.
More precisely he means that if $\phi:A \rightarrow Z$ is a morphism that there exists a unique function $\bar{\phi}$ such that $\bar{\phi}\pi = \phi$. However, I am having trouble conceptualizing this.
In particular, I am thinking of the example in which $Z = A$. Clearly, $1_A:A \rightarrow A$ exists as long as $A$ is non-empty. Also, a surjective canoncial projection $\pi:A \rightarrow A/\sim$ exists for any equivalence relation $\sim$ on $A$.
My problem then, is that the above theorem states that there exists some unique function $\bar{1_A}$ such that $\bar{1_A}\pi = 1_A$. However, this appears to imply that canonical projections have left-inverses, meaning they are injective!! This is obviously not true.
What mistake am I making??
The quotient $(\pi,A/\sim)$ is an initial object, not of the category whose objects are arbitrary maps out of $A$, but of the category whose objects are maps out of $A$ which identify elements of $A$ equivalent under $\sim$. So $1_A$ is an object in the relevant category if and only if $\sim$ is the discrete relation, in which case $\pi=1_A$ and there's no problem.