# A name and notation for a category

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 $f(a)=b$.

• How do you define composition? – Stefan Perko Nov 27 '16 at 9:51
• @StefanPerko as composition of functions – porton Nov 27 '16 at 16:33
• 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 $f\circ g$?) – Stefan Perko Nov 27 '16 at 16:41
• @StefanPerko $g\circ f$ is the function with domain $\{1\}$ which maps $1$ to $1$ – porton Nov 27 '16 at 16:44
• 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. – Stefan Perko Nov 27 '16 at 16:52