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Let $C$ a (perhaps well-pointed?) category with terminal object $\mathbf{1}$, so for objects $A,B$ we have the sets of global elements $G_A, G_B$ (i.e. hom-sets off $\mathbf{1}$) of $$ g_A \colon \mathbf{1} \to A, \qquad g_B \colon \mathbf{1} \to B. $$

Is the set of functions $G_A \to G_B$ isomorphic to the hom-set $\mathrm{Hom}(A,B)$?

In other words, is the function $f \colon \mathrm{Hom}(A,B) \times G_A \to G_B,$ $$ f(m, g_A) = m \circ g_A $$ bijective if considered curried, i.e. $\mathrm{Hom}(A,B) \to (G_A \to G_B)$?

I came up with this while reasoning about Are Monoids a category inside a category, wondering whether the endomorphism monoid is the same as the endofunction monoid on global elements.
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leftaroundabout
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    Are you asking if $\hom(A,B) \to \hom(\hom(1,A),\hom(1,B))$ is bijective, i.e. if $\hom(1,-)$ is fully faithful? Have you considered examples? $\mathsf{Ab}$ for instance (but this is not well-pointed, though). – Martin Brandenburg Feb 06 '14 at 14:09
  • @MartinBrandenburg: I did and still do think about examples, but I tend to lose track very quickly in anything less intuitive than $\mathrm{Set}$, $\mathrm{Top}$, or $\mathrm{Hask}$. – leftaroundabout Feb 06 '14 at 14:31
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    If you drop the well-pointedness requirement, its very, very false. For example, let $A$ denote a group with $|A|>2$. Then $A$ has a non-trivial automorphism. In other words, $\mathrm{hom}(A,A)$ has two or more elements. However, there is a unique arrow $1 \rightarrow A.$ – goblin GONE Feb 06 '14 at 14:47

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No. However, counterexamples are difficult to construct. For simplicity, assume we have a countable model $M$ of ZFC. Then we can construct the category of $M$-sets, and it will be well-pointed. Since $M$ is countable, all the hom-sets will also be countable. But there must exist an $M$-set $N$ with infinitely many (global) elements, and the set of all endomaps of an infinite set is always uncountable.

Zhen Lin
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