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Let f:a→b in the category of sets. Let f' be an object in the category of sets, which represents the homset containing only f.

Given that f and f' represent the same thing (just at different levels of abstraction), is there some standard notation which is used to explicitly link f to f' (and visa versa) in category theory?

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  • $\begingroup$ It's possible but not typical that $\mathsf{Hom}(a,b)=\{f\}$. At any rate, you can use the cartesian closure of $\mathbf{Set}$ to turn an arrow $1\to\mathsf{Hom}(a,b)$ into an arrow $a\to b$. This is essentially uncurrying (combined with simplifying $1\times a$ to $a$). This does not depend on $\mathsf{Hom}(a,b)$ being a singleton set. $\endgroup$ Jul 15, 2017 at 21:13
  • $\begingroup$ @DerekElkins So, there is no scenario (and hence no notation) for directly linking an object (for lack of better terms, 1-cat objects) to a function (2-cat objects)? I come from an engineering background, so when I look at diagrams such as in category theory I first ask "what else can I do". If its not possible, that is also fine. =) If you leave it as an answer then I'll mark it as correct. $\endgroup$ Jul 16, 2017 at 8:33

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