A question about filtered colimits in a category of representations

Question

For $k$ a field, are filtered colimits exact in the category $\mathbf{Rep}_k(G)$ of (finite-dimensional) $k$-representations of a group $G$? I can neither prove it nor find a counterexample.

Answer 1

For the category of all representations, filtered colimits are indeed exact because the forgetful functor from representations to vector spaces has adjoints on both sides; the left adjoint is $V\mapsto V\times G$ and the right, $V\mapsto \text{Hom}(k[G],V)$. Since the forgetful functor is monadic and preserves colimits, it creates all colimits (and limits). Since the forgetful functor is also conservative, exactness of filtered colimits follows from the same fact for vector spaces.