  Let $F$ be a local or global field and let $G$ be a linear algebraic group
over $F$. We study Tannakian categories of representations of the Kottwitz
gerbes $\text{Rep}(\text{Kt}_{F})$ and the functor $G\mapsto B(F, G)$ defined
by Kottwitz in [28]. In particular, we show that if $F$ is a function field of
a curve over $\mathbb{F}_q$, then $\text{Rep}(\text{Kt}_F)$ is equivalent to
the category of Drinfeld isoshtukas. In the case of number fields, we establish
the existence of various fiber functors on $\text{Rep}(\text{Kt}_{\mathbb{Q}})$
and its subcategories and show that Scholze's conjecture [41, Conjecture 9.5]
follows from the full Tate conjecture over finite fields [47].
