# 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 $$f' \circ a = b \circ f$$

• Consider what the final objects in $\mathbf{Set}$ are; this should give you a clue as to what the final objects in this category should be. – Clive Newstead Aug 19 '19 at 20:27
• If we write $\to$ for the category consisting of two objects and one arrow joining them (and the identities), then the arrow category of $\mathcal C$ is $\mathcal C^{\to}$, i.e. the category of functors from $\to$ to $\mathcal C$. Now you can apply generic theorems about limits in functor categories, in particular that they are computed point-wise when the target category is complete. You are probably more at a point where you should just try to directly prove the statement though... – Derek Elkins left SE Aug 19 '19 at 21:00

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?