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These thoughts at least helped me to understand the connection between models and clones as defined in UA more precisely. I, in fact, like to think of clones as I have described above, so this would be my suggestion how to present them. However, I can hardly clame that this is the best way - it depends on what you want and, of course, your personal taste. In fact, what I find most interesting is to find categories $\mathcal{C}$ and an object $\mathbf{A} \in \mathcal{C}$ such that the clone $O_{\mathbf{A}}$ is equivalent to a certain clone in $Set$, but can now be investigated differently since the categorical environment has changed. This might give universal algebraist some results that would have been much harder to get without the aforementioned change in the environment. In particular, applying duality is then possible and might be interesting (as I have tried to describe in the thread Lawvere theories versus classical universal algebra)

These thoughts at least helped me to understand the connection between models and clones as defined in UA more precisely. I, in fact, like to think of clones as I have described above, so this would be my suggestion how to present them. However, I can hardly clame that this is the best way - it depends on what you want and, of course, your personal taste. In fact, what I find most interesting is to find categories $\mathcal{C}$ and an object $\mathbf{A} \in \mathcal{C}$ such that the clone $O_{\mathbf{A}}$ is equivalent to a certain clone in $Set$, but can now be investigated differently since the categorical environment has changed. This might give universal algebraist some results that would have been much harder to get without the aforementioned change in the environment. In particular, applying duality is then possible and might be interesting (as I have tried to describe in the thread Lawvere theories versus classical universal algebra)

Ich think you should not compare Lawvere theories with clones from universal algebra, as the two things are not on the same level of generality even within their respective field. What, in my oppinion, you should compare are

I think you should not compare Lawvere theories with clones from universal algebra, as the two things are not on the same level of generality even within their respective field. What, in my oppinion, you should compare are

Cearly, this definition is just the classical definition of a clone if $\mathcal{C} = Set$. It is also "essentially" a model of a Lawvere theory. By that, I mean the following: A set of morphisms $C$ is a clone over the object $\mathbf{A}$ in the sense given above if any only if there exists a Lawvere theory $\mathcal{L}$ with objects $\mathbf{t_0},\mathbf{t_1},\ldots$ (where $\mathbf{t_i}$ is the $i$-th power of $\mathbf{t_1}$) and a model $M \colon \mathcal{L} \rightarrow \mathcal{C}$ such that $\mathbf{A} = M(\mathbf{t_1})$ and $C = \{ M(f) \mid f \in \mathcal{L}(\mathbf{t_n},\mathbf{t_1}) \}$.

Cearly, this definition is just the classical definition of a clone if $\mathcal{C} = Set$ (except that the nullary operations are often excluded from the classical definition, but please let us not get into that). It is also "essentially" a model of a Lawvere theory. By that, I mean the following: A set of morphisms $C$ is a clone over the object $\mathbf{A}$ in the sense given above if any only if there exists a Lawvere theory $\mathcal{L}$ with objects $\mathbf{t_0},\mathbf{t_1},\ldots$ (where $\mathbf{t_i}$ is the $i$-th power of $\mathbf{t_1}$) and a model $M \colon \mathcal{L} \rightarrow \mathcal{C}$ such that $\mathbf{A} = M(\mathbf{t_1})$ and $C = \{ M(f) \mid f \in \mathcal{L}(\mathbf{t_n},\mathbf{t_1}) \}$.