To state my question, I first have to explain what I mean by "using operations and their compositions only". In short, it means that I am looking for theorems from clone theory (or on operations and relations in general) that can still be stated if we consider the operations as morphisms in an abstract category that has products (why we need the products will be explained below).

I believe that this can be best explained by using an example.

Take, for instance the well-known Swierczkowski-Lemma that says that if you obtain a projection from a given at least quaternary operation by identifying two of its arguments, then this projection must always be the same no matter which two variables one identifies. This lemma can be stated only be using operations and compositions between them. This can be done as follows.

Let $A$ be an object in a category with products. Let $f$ be a morphism from $A^n$ to $A$ ($n \geq 4$) such that $\langle \pi_{i_1}^n,\ldots,\pi_{i_n}^n \rangle \circ f$ is a projection whenever $i_1,\ldots,i_n$ are not pairwise distinct (of course, the projections are all defined from $A^n$ to $A$). Then, the projection is always the same.

In the category of sets, this Lemma coincides with the usual Swierczwoski-Lemma. The only things we needed for this statement are the operations (interpreted as morphisms from the $n$-th power of $A$ to $A$), the composition between them and the fact that we have products (such that the powers of $A$ and hence also the tuplings and the projections are available).

So my question is: What other theorem (or lemmas) from clone theory do you know that can be stated by only using these things? Note that I am not asking for the theorem to be still correct. This is something I will then figure out on my own (as a sidenote: the above stated "generalized" Swierczwoski-Lemma is still true). I just want to find candidates.