Is there a name and a standard notation for the category $C$ defined as follows:

  • objects of $C$ are all small "objects" (small sets in ZF, because in ZF all objects are sets);

  • morphisms of $C$ from $a$ to $b$ are all small functions such that $f(a)=b$.

  • $\begingroup$ How do you define composition? $\endgroup$ – Stefan Perko Nov 27 '16 at 9:51
  • $\begingroup$ @StefanPerko as composition of functions $\endgroup$ – porton Nov 27 '16 at 16:33
  • $\begingroup$ How do you compose the function $f : \{1\}\to \{1\},x \mapsto x$, which is a morphism $1\to 1$ with the function $g : \mathbb{R}\to \mathbb{R}, x\mapsto x^2$, which is a morphism $1\to 1$? (What is $f\circ g$?) $\endgroup$ – Stefan Perko Nov 27 '16 at 16:41
  • $\begingroup$ @StefanPerko $g\circ f$ is the function with domain $\{1\}$ which maps $1$ to $1$ $\endgroup$ – porton Nov 27 '16 at 16:44
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    $\begingroup$ I still (personally) don't understand how composition is defined. Saying "function composition" is not enough since there is no reason why two of these morphims need to be composable. $\endgroup$ – Stefan Perko Nov 27 '16 at 16:52

I would probably call this something like "the subcategory of epimorphisms in Set". If I were to use it a lot, especially with other categories, I would probably name it something like Epi(Set).

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