In the case of a dyadic relation 
![\begin{array}{lll} f ~\text{is surjective} & \iff & f ~\text{is total at}~ Y. \\[6pt] f ~\text{is injective} & \iff & f ~\text{is tubular at}~ Y. \\[6pt] f ~\text{is bijective} & \iff & f ~\text{is}~ 1\text{-regular at}~ Y. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++f+%7E%5Ctext%7Bis+surjective%7D+%26+%5Ciff+%26+f+%7E%5Ctext%7Bis+total+at%7D%7E+Y.++%5C%5C%5B6pt%5D++f+%7E%5Ctext%7Bis+injective%7D++%26+%5Ciff+%26+f+%7E%5Ctext%7Bis+tubular+at%7D%7E+Y.++%5C%5C%5B6pt%5D++f+%7E%5Ctext%7Bis+bijective%7D++%26+%5Ciff+%26+f+%7E%5Ctext%7Bis%7D%7E+1%5Ctext%7B-regular+at%7D%7E+Y.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002 "\begin{array}{lll} f ~\text{is surjective} & \iff & f ~\text{is total at}~ Y. \\[6pt] f ~\text{is injective} & \iff & f ~\text{is tubular at}~ Y. \\[6pt] f ~\text{is bijective} & \iff & f ~\text{is}~ 1\text{-regular at}~ Y. \end{array}")
For example, the function 
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(40) |
An easy way to extract a surjective function from any function is to reset its codomain to its range. For example, the range of the function 
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(41) |
The function 
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(42) |
The function 
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(43) |
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