CatDat

Exact filtered colimits descend to nice subcategories

Proof. It is well-known (and easy to prove) that the colimit of a diagram $(X_j)$ in $\C$ is constructed as $F(\colim_j G(X_j))$, provided that colimit in $\D$ exists. In particular, $\C$ has filtered colimits. By assumption, it also has finite limits, and $G$ preserves these since it is a right adjoint. Now let $X : \I \times \J \to \C$ be a diagram, where $\I$ is finite and $\J$ is filtered. We compute:

$$\begin{align*} \colim_j {\lim}_i X(i,j) & \cong F(\colim_j G({\lim}_i X(i,j))) \\ & \cong F(\colim_j {\lim}_i G(X(i,j))) \\ & \cong F({\lim}_i \colim_j G(X(i,j))) \\ & \cong {\lim}_i F(\colim_j G(X(i,j))) \\ & \cong {\lim}_i \colim_j X(i,j) \end{align*}$$

$\square$