  We associate to every $G$-bornological coarse space $X$ and every left-exact
$\infty$-category with $G$-action a left-exact infinity-category of equivariant
$X$-controlled objects. Postcomposing with algebraic K-theory leads to {new}
equivariant coarse homology theories. This allows us to apply the injectivity
results for assembly maps by Bunke, Engel, Kasprowski and Winges to the
algebraic K-theory of left-exact $\infty$-categories.
