I read the statement that "a clonal category has a small set of objects", which I don't quite agree about. In the definition of clonal category, at least as it is given in that context, it is required that a bijection exist between its object-set and the set of natural numbers $\omega$. This, in my understanding, does not imply that this object-set is small. Do you agree or am I missing something? Thanks in advance for any answer of yours.
PS Yes, I did miss something. My oversight was that throughout the text the hypothesis is tacitly understood, that, though a category may have a large set of objects, these are always small, and of course a countable set of small sets is indeed small. Thanks to everybody though.
The set $\omega$ is surely small.
And every set in bijection with a small set is small (working in ZF for example, it is a special case of the replacement axiom scheme).