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Is there a characterization in the literature of the Lawvere theories that correspond to (algebras in Set for) operads (in Set)?
I presume you mean nonsymmetric operads rather than symmetric ones? Symmetric operads are treated in Szawiel–Zawadowski's Theories of analytic monads, but I haven't come across a similar result for nonsymmetric operads.
I am mainly interested in symmetric operads, so this looks like exactly what I wanted (at least in the 1-sorted case), thanks!