Let $K$ be an algebraically closed field and consider the category $\text{Bun}$ of (finite dimensional) vector bundles over a $K$-variety $X$. Consider also the category of $\mathbb{G}^\times$-principal bundles over $X$, $\text{Bun}_0$. $\mathbb{G}^\times$-principal bundles are equivalent to maps into $\mathbb{P}^{\infty}$, and with this I have a much clearer picture of how $\text{Bun}_0$ works in algebraic geometry e.g. with the clutching construction.
However, the cohomology classes in algebraic geometry are more typically understood as being valued in $\text{Bun}$ (not just $\text{Bun}_0$). Somehow it is possible to extend from the case for line bundles so long as $l_1 \oplus \cdots \oplus l_n$ is sent to $\text{cl} (l_1) \cup \cdots \cup \text{cl}(l_n)$. I see of course that such an extension is unique when it exists, but I do not as of yet understand why we can consistently extend maps from $\text{Bun}_0$ to $\text{Bun}$.
So my question is, what is an extension theorem that goes something like this (ideally with a reference):
A natural transformation from $\eta : \text{Bun}_0 = \text{Bun}_0(X) \implies F$ between contra variant functors on $K$-varieties always extends to a natural transformation $M(\eta) : \text{Bun} \implies F$ preserving the apparent product structure given by $* \rightarrow X \leftarrow X \times X$.
