Final Objects of a (Coslice?) Category I'm reading Aluffi's Algebra: Chapter 0 and in example 5.6 / exercise 5.5, he asks what the terminal object/objects (he refers to it as singular in the example but in plural in the exercise) are the in following category. He also remarks that its a supremely uninteresting one.
The category is defined as such:
Given an equivalence relation $\sim$ on set $A$,
Objects of the category are morphisms $A \overset{\varphi}{\longrightarrow} Z$  from the set $A$ to any set $Z$ such that $\alpha \sim \alpha' \Rightarrow \varphi(\alpha) = \varphi(\alpha')$ (I think this is an example of a coslice category).
Morphisms between objects $(\varphi_1,Z_1) \rightarrow (\varphi_2,Z_2)$ are defined as morphisms $\sigma$ such that the following commutes (can't draw a commutative diagram with diagonal arrows in mathjax so imagine $A$ being the bottom of a triangle with the other two vertices being $Z_1$ and $Z_2$) :
$$ A \overset{\varphi_1}{\longrightarrow} Ζ_1  \overset{\sigma}{\longrightarrow} Z_2 \overset{\varphi_2}{\longleftarrow} A $$
So far I've figured that both the morphism $(1_A, A)$ (that is the identity morphism from $A$ to $A$) and the morphisms of type $(const_Α, S)$ (where $S$ is any singleton set and $const_A$ is the constant function from $A$ to that set) appear to be final objects in this category.
The unique morphisms from any $(\varphi, Z)$ to $(1_A , A)$ are the reverse functions $\varphi^{-1}$ while the unique morphism for any from any $(\varphi, Z)$ to $(const_Α, S)$ are the corresponding constant functions from $Z$ to the singleton $S$.
Is this correct? Any help is greatly appreciated!
 A: There are some problems with your answer.
First of all, a terminal object is unique up to (unique) isomorphism. Hence you cannot have two non isomorphic terminal objects.
You are right that the singleton set with the constant map is a terminal object. (And every terminal object is a singleton set).
However, $(id_A,A)$ is not a terminal object. In fact, it is not an object at all unless the equivalence relation $\sim$ is the identity. Remember that an object of your category is a map $\varphi:A\rightarrow Z$ such that $\varphi(a)=\varphi(a')$ if $a\sim a'$. The identity satisfies this condition if and only if $\sim$ is the equality.
But even in that case, it is not the terminal object. You said, the unique function form $(\varphi,Z)$ to $(id_A,A)$ is the reverse function $\varphi^{-1}$. But it does not exist if $\varphi$ is not bijective. Can you show that there is no map from $(const,\{*\})$ to $(id_A,A)$ ?
But for this category, this is the initial object that is very interesting. Can you see what it is ? Hint : it is a usual construction given a set equipped with an equivalence relation.
