Pullbacks in the Ind-completion Suppose we have a category $\mathcal{C}$, say finitely complete. Does the $\text{Ind}$-completion $\text{Ind}(\mathcal{C})$ (which informally is the completion of $\mathcal{C}$ under filtered colimits -- see here for a precise construction) have pullbacks? Can these be described (relatively) concretely?
 A: If $\mathcal{C}$ has finite limits then $\mathbf{Ind}(\mathcal{C})$ has finite limits. (However, the converse is false.)
Indeed, let $\mathcal{J}$ be a finite diagram. By hypothesis, there is an adjunction
$$\Delta \dashv \varprojlim : [\mathcal{J}, \mathcal{C}] \to \mathcal{C}$$
where $\Delta : \mathcal{C} \to [\mathcal{J}, \mathcal{C}]$ sends each object in $\mathcal{C}$ to the corresponding constant diagram of shape $\mathcal{J}$. Thus, there is an induced adjunction
$$\mathbf{Ind}(\Delta) \dashv \mathbf{Ind}(\varprojlim) : \mathbf{Ind}([\mathcal{J}, \mathcal{C}]) \to \mathbf{Ind}(\mathcal{C})$$
and by using the fact that filtered colimits preserve finite limits in $\mathbf{Set}$, it can be shown that the canonical functor
$$\mathbf{Ind}([\mathcal{J}, \mathcal{C}]) \to [\mathcal{J}, \mathbf{Ind}(\mathcal{C})]$$
is fully faithful and essentially surjective on objects. Thus, $\Delta : \mathbf{Ind}(\mathcal{C}) \to [\mathcal{J}, \mathbf{Ind}(\mathcal{C})]$ has a right adjoint, i.e. $\mathbf{Ind}(\mathcal{C})$ has limits for diagrams of shape $\mathcal{J}$.
Unfortunately, it is not easy to give a concrete description of finite limits in $\mathbf{Ind}(\mathcal{C})$ – the hard part is in describing the equivalence $\mathbf{Ind}([\mathcal{J}, \mathcal{C}]) \to [\mathcal{J}, \mathbf{Ind}(\mathcal{C})]$ explicitly.
