There is a coloured operad $sOp$ such that $sOp$-algebras are single-coloured operads. This operad has a simple description in terms of generators and relations, say, as an operad $F(X)/R$. There is a more concrete description of $sOp$ as the operad $OpTrees$ of trees endowed with additional structure, see [1, section 4] or [2, Example 1.5.6]. The proof that the operads $F(X)/R$ and $OpTrees$ are isomorphic is not too hard, but not entirely trivial either.


Likewise, there should be a cartesian multicategory (or equivalently a multi-sorted Lawvere theory, or a multi-sorted abstract clone) $CartOp$ such that $CartOp$-algebras are single-coloured cartesian multicategories. Definition of $CartOp$ in terms of generators and relations seems fairly straightforward. However, concrete description of $CartOp$, description similar to that of $OpTrees$, is a bit more elusive.


Is there any reference that gives concrete description of $CartOp$?


[1] Pepijn van der Laan, Coloured Koszul duality and strongly homotopy operads <https://arxiv.org/abs/math/0312147>


[2] Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras <https://arxiv.org/abs/math/0512576>




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Update. There is "Operads from the viewpoint of categorical algebra" of Tibor Beke, which gives description in terms of monads. I'll close the question. Any reference with a clean proof of isomorphism analogous to that of $F(X)/R\cong OpTrees$ is still welcome.



Lawvere theory of Lawvere theories
Viewing cartesian operads as algebraic theories, your question may be rephrased as: "How can we present the (multisorted) algebraic theory whose models are (monosorted) algebraic theories?"


This is given by the theory of [abstract clones](https://en.wikipedia.org/wiki/Clone_(algebra)#Abstract_clones). You can check that the category of monosorted algebraic theories is isomorphic to the category of abstract clones, the latter of which is defined as an $\mathbb N$-sorted universal algebra. It is then straightforward to translate back from algebraic theories to cartesian multicategories / cartesian coloured operads.


