What are the final objects of the $Arrow(Set)$ category? By $Arrow(Set)$ I mean the category whose objects are the arrows of the Set category and whose arrows from the object $f : A \rightarrow B$ to the object $f' : A' \rightarrow B'$ are the pairs of functions $(a,b)$ such that  $f' \circ a = b \circ f$
 A: I have tried to give you a way of seeing what the answer should be (without telling you precisely).
To find a terminal (final) object we want a map $t: S\to T$ such that for all maps $f: A\to B$ there exist unique maps $a :A \to S$ and $b: B\to T$ such that $ta=bf$. For convenience lets write such a morphism as $(a,b):f\to t$. Suppose such a $t$ exists. Now for any two maps $u : W\to S$  and $v : W\to S$ we obtain morphisms $(u,tu): 1_W \to t$ and $(v,tv) : 1_W \to t$. What can we conclude about $(u,tu)$ and $(v,tv)$ and hence about $u$ and $v$? On the other hand for any set $C$ there must be a morphism from $1_C$ to $t$ and hence a map from $C$ to $S$. What does this and the previous part tell us about $S$? After that let $(x,y) : 1_T\to t$ be the unique morphism and note that $(xt,y)$ is a morphism from $t$ to $t$ and hence must be $(1_S,1_T)$ meaning that $xt=1_S$ and $y=1_T$. This means that $txt=t$ and hence $(1_S,tx)$ is a morphism from $t$ to $t$. Why does this mean that $tx=1_T$? What can you conclude?
