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Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?

Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set

Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?

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Small set of acts over a countable monoid?

Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set