Indization as adjoint from finite colimits to all colimits Where can I find a reference for the following fact:
If $C$ is a small category with finite colimits, then $\text{Ind}(C)$ is a category with all colimits, and it is the universal one in the following sense: If $D$ is a category with all colimits, then there is an equivalence of categories between finitely cocontinuous functors $C \to D$ and cocontinuous functors $\text{Ind}(C) \to D$.
This is a consequence of a combination of facts which can be found in SGA 4 I.8 and Section 6 in "Category of Sheaves" by Kashiwara, Shapira. I wonder if this is written down somewhere so that I can reference it in my work without spelling out the proof.
EDIT: Actually I don't know exactly how to prove it. Take a finitely cocont. functor $C \to D$. This extends to a cocont. functor $\widehat{C} \to D$ and may be restricted to $\text{Ind}(C)$. The resulting functor $\text{Ind}(C) \to D$ commutes with coproducts and with filtered colimits. The problem are the cokernels (or rather coequalizer, if we do not deal with linear categories). Namely, we should prove that every cokernel diagram in $\text{Ind}(C)$ is a filtered colimit of cokernel diagrams in $C$. Now in the book by Kashiwara, Schapira it is shown that (almost) every finite diagram in $\text{Ind}(C)$ is a filtered colimit of diagrams of the same shape in $C$. But we need a refinement of this!
 A: This is not what you asked (again!) but the derived ($\infty$-categorical) version of this statement is Proposition 5.3.5.10 in Lurie's Higher Topos Theory.
A: This book has a pretty thorough discussion of Ind-completion:
\bib{MR861951}{book}{
   author={Johnstone, Peter T.},
   title={Stone spaces},
   series={Cambridge Studies in Advanced Mathematics},
   volume={3},
   note={Reprint of the 1982 edition},
   publisher={Cambridge University Press},
   place={Cambridge},
   date={1986},
   pages={xxii+370},
   isbn={0-521-33779-8},
   review={\MR{861951 (87m:54001)}},
}

However, I do not have my copy to hand, so I cannot guarantee that it contains precisely the result that you want.
A: Every finite loopless diagram in Ind-$C$ can be represented as a levelwise diagram, i.e. as a diagram of functors $I \to C$ for some filtered $I$. (Artin-Mazur, App. of Etale Homotopy Theory, Prop. 3.3 for example.) In particular, this holds for coequalizer diagrams. Then the colimit can be computed levelwise (Prop. 4.1 in Artin-Mazur). See also D. Isaksen, "Calculating limits and colimits in pro-categorie", Fund. Math. 175 (2002).
A: If your definition of $\operatorname{Ind}(C)$ is "those ($\operatorname{SET}$-valued) presheaves on $C$ that take finite colimits to limits in $\operatorname{SET}$" (so that it is a full subcategory of $\{\text{presheaves on }C\}$), then the fact follows from general facts about presentable categories, and a good reference is Adámek and Rosický, Locally presentable and accessible categories, 1994.  But this probably is not your definition of $\operatorname{Ind}(C)$, and so you would have to prove it is equivalent, and by then you have probably proved the result you want.
