I know that the category of groups $\textbf{(Group)}$ and rings $\textbf{(Ring)}$ are both only locally small, since any non-empty set can be made into a group or a ring.
However, when this comes to fields, I'm not sure if there's also an explicit construction making an infinite set to a field (we know not all finite set can be a field). Therefore I cannot confirm if the category of fields is small.
Thanks for any suggestion in advance.
Let's fix the rational numbers. Right? That's just one set now. Given any other set which is disjoint from $\Bbb Q$, consider the transcendental extension $\Bbb Q(A)$, as a field. This alone shows that every set can be embedded into a field. And easy cardinality check will show that $|A|=|\Bbb Q(A)|$ whenever $A$ is infinite.
In particular, via transport of structure any infinite set can be made into a field. And so far we're only talking about purely transcendental extensions of $\Bbb Q$. This can be repeated with any field instead of $\Bbb Q$ (e.g. $\Bbb F_2$) as well.