Characterization of Schur Functors A Schur functor (in the theory of algebraic operads) on the category of vector spaces over a field $k$,(or more generally any abelian symmetric tensor category) is defined by:
$$\tilde{M}(V) := \bigoplus_{n\geq 0} M(n)\otimes_{S_n} V^{\otimes n}$$
($M(n)$ is a vector space with right $S_n$ action for every $n$) 
Does there exists a categorical characterization  of such endofunctors of $\mathrm{Vect}_k$ (as an abelian tensor category)? e.g. if $M$ is supported in degree $1$, (i.e. $M(n)=0$ for every $n\neq 1$) then the resulting functors are exactly the additive right exact endofunctors. Does there exists a similar interpretation in other cases?
Thanks!
 A: Here is one thing you can say, although it may not be the sort of result you have in mind. First of all, it does not suffice that your tensor category be abelian; if $M$ isn't finitely supported you also need infinite coproducts. It is better to work with symmetric monoidal cocomplete $k$-linear categories (this includes the condition that the monoidal operation is $k$-bilinear and preserves colimits in both variables). 
In this setting Schur functors naturally arise as follows. The free symmetric monoidal cocomplete $k$-linear category on an object turns out to be the category of "$k$-linear species," or equivalently the category of $\text{Vect}_k$-valued presheaves on $\text{FinSet}^{\times}$, the groupoid of finite sets and bijections, or equivalently the category of sequences $M(n)$ of $k$-vector spaces equipped with right actions of the symmetric groups $S_n$. $\text{FinSet}^{\times}$ itself is the free symmetric monoidal category on an object, with symmetric monoidal structure given by disjoint union, and this symmetric monoidal structure induces a symmetric monoidal structure on $k$-linear species by Day convolution. 
In particular, this universal property states that every object $V$ in a symmetric monoidal cocomplete $k$-linear category $C$ naturally gives rise to a functor from $k$-linear species into $C$, and this functor is given precisely by
$$M(n) \mapsto \bigoplus_{n \ge 0} M(n) \otimes_{S_n} V^{\otimes n}.$$
The universal property furthermore states that every symmetric monoidal cocontinuous $k$-linear functor from $k$-linear species to $C$ has this form, and in fact that the category of such functors is equivalent to $C$. In particular, taking $C$ to be $k$-linear species itself, we see that the symmetric monoidal cocontinuous $k$-linear endofunctors of $k$-linear species can be identified with $k$-linear species, but with a new monoidal structure corresponding to composition of Schur functors.
In particular, taking $C$ to be $k$-linear species itself and using a suitable version of the Yoneda lemma, we see that Schur functors are precisely the natural unary operations on symmetric monoidal cocomplete $k$-linear categories in the sense that they are precisely the natural endomorphisms of the forgetful functor from symmetric monoidal cocomplete $k$-linear categories to categories. See this blog post for some examples, with categories replaced by sets, of why this construction deserves to be called "unary operations."
Among other things, this point of view gives a nice characterization of $k$-linear operads: they are precisely the monoid objects in $k$-linear species with respect to the composition monoidal structure, which means that they are precisely the "natural monads" on symmetric monoidal cocomplete $k$-linear categories. 
