I'm kinda lost at one example in Awodey's: Category Theory.
I am trying to check this example, for that, I made a simple example function:
With this, I mean that there is one function $f$ which associates each element of the codomain. In this case, I can't have an inverse function - I'm not sure if this invalidates it being a category, but I faintly believe it does. Because it seems that in this universe of discourse, the arrows are functions. But I guess I could counter this by creating two pseudo-inverse functions (?!) in the following way:
It doesn't seem to violate the rules, except a little doubt for one of them: The definitions of composition. If I compose $f_{1}^{-1} \circ f$, I have:
$$cod(f)=dom(f_{1}^{-1})$$
$$cod(f_{1}^{-1})=dom(f)$$
I believe this is valid because he speaks about codomain instead of image. But it feels weird, $f_{1}^{-1}$ doesn't goes back entirely to $dom(f)$, that is: It doesn't takes all the elements of $dom(f)$.
Or perhaps I'm utterly confused/lost/stupid/in_need_of_help_for_mental_illness and completely missed the point.