In *[Rethinking set theory](http://arxiv.org/pdf/1212.6543v1.pdf)*, in which an axiomatization of set theory called ETCS is introduced, Tom Leinster writes:

> 
> First we state the data to which our axioms
>  will apply:  
> 
>  • Some things called sets;  
> 
>  • for each set X and set Y , some things called
>  functions from X to Y
> 
> 
> 

So, in ETCS there are some sorts (sometimes also called "types"). One sort is set.

In *[fully formal ETCS](http://ncatlab.org/nlab/show/fully+formal+ETCS)*, Todd Trimble writes:

> 
> The usual presentation [of ETCS] involves two sorts, objects and morphisms.
> 
> 
> 

Is it right that ETCS, then, has only the two types "set" and "function"?
Or, to be precise, is there a type "function from A to B" for **each** sets A, B? That is: Is there (for A, B sets) a separate type "function from A to B"?
Or, to put it in other words: Does ETCS has a infinite number of types or a finite number of types?