Denote by $Vect^{fin}\_{\mathbb{C}}$ the category of finite dimensional complex vector spaces. We will call a pair $(F,\tau)$ that consists of a functor $F \colon Vect^{fin}\_{\mathbb{C}} \to Vect^{fin}\_{\mathbb{C}}$ and a natural isomorphism 
$$
\tau\_{V,W} \colon F(V \oplus W) \to F(V) \otimes F(W)
$$ 
an *exponential functor* if $\tau$ satisfies the following associativity condition: 
$$
(\tau\_{V,W} \otimes id\_{F(X)}) \circ \tau\_{V \oplus W, X} = (id\_{F(V)} \otimes \tau\_{W,X}) \circ\tau\_{V,W \oplus X}
$$
An example of such a functor is the full exterior power $\Lambda^\*$ together with the natural iso $\Lambda^\*(V \oplus W) \cong \Lambda^\*(V) \otimes \Lambda^\*(W)$. Another example along the same lines is 
$$
F\_X(V) = \Lambda^\*(V \otimes X)
$$
for a fixed finite dimensional complex vector space $X$. Since we insist on finite dimensional vector spaces as our target category for $F$, the full symmetric power is not an example.



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exponential functors on finite dimensional complex vector spaces
In case anyone is still interested in this question: 


There is a classification of *polynomial* exponential functors on the category $\mathcal{V}$ of finite-dimensional inner product spaces in terms of involutive $R$-matrices (i.e. involutive solutions to the Yang-Baxter equation). Basics about polynomial functors can be found in the book "[Symmetric Functions and Hall Polynomials](https://books.google.co.uk/books?id=srv90XiUbZoC&printsec=frontcover&dq=isbn:0198504500&hl=en&sa=X&ved=0ahUKEwiZk5_6wsjaAhVRbFAKHfaGDcUQ6AEIKTAA#v=onepage&q&f=false)" by Macdonald (Chapter 1, Appendix A).


A functor $F \colon \mathcal{V} \to \mathcal{V}$ is polynomial, if for all linear maps $f\_i \colon V \to W$ the result of $F(\lambda\_1 f\_1+ \dots + \lambda\_n f\_n)$ is a polynomial in the $\lambda\_i$ with coefficients in $\hom(V,W)$. Each such functor has a direct sum decomposition into homogeneous components $F\_n \colon \mathcal{V} \to \mathcal{V}$. Each homogeneous component has a linearisation $L\_{F\_n} \colon \mathcal{V}^n \to \mathcal{V}$ and $L\_{F\_n}(V\_1, \dots, V\_n)$ is defined to be the direct summand of $F\_n(V\_1 \oplus \dots \oplus V\_n)$, on which $\lambda\_1 id\_{V\_1} \oplus \dots \oplus \lambda\_n id\_{V\_n}$ acts via multiplication by $\lambda\_1 \cdot \lambda\_2 \cdots \lambda\_n$.


For an exponential functor $F$, the linearisation $L\_{F\_n}$ of its $n$-homogeneous summand $F\_n$ is equivalent to $F\_1^{\otimes n}$. Let $W = F\_1(\mathbb{C})$ and $\tau \colon \mathbb{C}^2 \to \mathbb{C}^2$ be the isomorphism permuting the summands. Then $F\_2(\tau) \colon F\_2(\mathbb{C}^2) \to F\_2(\mathbb{C}^2)$ restricts to a linear transformation $R \colon W^{\otimes 2} \to W^{\otimes 2}$ of the linearisation of $F\_2$ that is involutive and satisfies the Yang-Baxter equation on $W^{\otimes 3}$. This can be extended to the statement that polynomial exponential functors up to natural equivalence of monoidal functors are in $1:1$-correspondence with certain equivalence classes of involutive $R$-matrices. 


This is written up [here](https://arxiv.org/abs/1804.05807). This result is used there to construct twists of complex $K$-theory localised at an integer over $SU(n)$, which was my original motivation for this question.


