Detecting Universals An operation on a category $C$ is a functor $$F: C^n \to C.$$
I'd like to detect those construction that are universal. Easiest example is coproduct, which is a colimit.
In this sense to be universal means that there is a small category $I$ and a functor $$ Q : C^n \times I \to C $$ such that $$\text{colim} Q(c_1, \dots, c_n) = F(c_1, \dots, c_n),$$ equivalently, one is looking for a lifting closing the following diagram:
$\require{AMScd}$
\begin{CD}
     && C^I\\
    & @VV \text{colim} V\\
    C^n @>>F> C
\end{CD}
And this is linked to find the suitable Kan Lift of this diagram, but I cannot speak Kan Lift language.
 A: I think you are not asking the right question. A distinguishing feature of the coproduct as an operation $C^n \to C$ (on categories with coproducts) is that it is (lax) natural in $C$: given a functor $F : C \to D$ where $D$ also has coproducts we get an induced diagram which lax (oplax?) commutes in the sense that there's a natural transformation
$$F(c_1) \sqcup \dots \sqcup F(c_n) \to F(c_1 \sqcup \dots \sqcup c_n)$$
which we can ask to be an isomorphism if we want $F$ to preserve coproducts. This is the behavior that really distinguishes coproducts from just a random functor $C^n \to C$ which may not "generalize" to other categories besides $C$ in any principled way. 
I don't know how to answer the question of how to classify operations on categories in this lax sense, although I will note that it obviously depends on what class of categories you consider (e.g. all categories, categories with coproducts, cocomplete categories, etc). I do know something about how to classify operations on categories in a non-lax sense. 
The idea is this: given some class $\text{MyCat}$ of categories (with extra properties or structure as desired), natural operations $C^n \to C$ correspond to natural transformations from the functor $C^n$, regarded as a functor from $\text{MyCat}$ to $\text{Cat}$, to the functor $C$, regarded as the forgetful functor from $\text{MyCat}$ to $\text{Cat}$. These functors are frequently representable by coproducts of a single category, namely the free category on a point in $\text{MyCat}$, so by the Yoneda lemma, the category of operations $C^n \to C$ can be identified with the free category on $n$ points in $\text{MyCat}$. For examples of the same idea one category level down, applied to sets with extra structure, see this blog post. Some examples with categories:


*

*If $\text{MyCat}$ is just $\text{Cat}$, then the free category on $n$ points is the discrete category on $n$ points, so the only operations $C^n \to C$ are the $n$ projection operations.

*If $\text{MyCat}$ is categories with finite coproducts, then the free category on $n$ points is $\text{FinSet}^n$. The operations $C^n \to C$ correspond to taking "linear combinations" of $n$ objects in $C$ with "coefficients" in $\text{FinSet}$. 

*If $\text{MyCat}$ is symmetric monoidal categories, then the free category on $n$ points is $\left( \text{FinSet}^{\times} \right)^n$, where $\text{FinSet}^{\times}$ denotes the symmetric monoidal category of finite sets and bijections, under componentwise disjoint union. The operations $C^n \to C$ are the same as above but with fewer natural transformations between them.

*If $\text{MyCat}$ is cocomplete categories, then the free category on $n$ points is $\text{Set}^n$. The operations $C^n \to C$ correspond to taking linear combinations of $n$ objects in $C$ with coefficients in $\text{Set}$. 

*If $\text{MyCat}$ is symmetric monoidal cocomplete categories (including the hypothesis that the monoidal structure distributes over colimits in both variables), then the free category on $n$ points is the category of presheaves on $(\text{FinSet}^{\times})^n$ under Day convolution; for $n = 1$ this is the category of species and in general it's a multivariate generalization of species. The operations $C^n \to C$ correspond to "formal exponential power series" in $n$ variables with coefficients in $\text{Set}$; I can write this out in more detail if desired. 

A: Every functor $F:C^n \to C$ is universal in your sense. Take $I$ to be any connected category and set $Q = F \circ \pi$ where $\pi : C^n \times I \to C^n$ is the projection.
