Do direct sums commute with colimits?

Question

Is it true that finite direct sums commute with colimits? That is, do we have $\varinjlim_{V\subset U} (A(U)\oplus B(U)) = (\varinjlim_{V\subset U} A(U))\oplus (\varinjlim_{V\subset U}B(U))$ where $A$ and $B$ are functors?

Answer 1

It is true, and indeed it's true for arbitrary direct sums, not just finite ones; moreover, it's true for arbitrary colimits, not just direct sums.

Colimits always commute with colimits. Indeed, if $D: I \times J \to \mathcal{C}$ is a diagram, where $\mathcal{C}$ has limits of shape $I$ and shape $J$, then both $\lim_I \lim_J D$ and $\lim_J \lim_I D$ satisfy the universal property of $\lim_{I \times J} D$. In particular, they are isomorphic. The dual of this result is what you want.