I suspect that this paper does what you need:

\bib{MR569571}{article}{
author={Kaphengst, H.},
author={Reichel, H.},
title={Finite limit-colimit completions of small categories},
conference={
title={Algebraische Modelle, Kategorien und Gruppoide},
},
book={
series={Stud. Algebra Anwendungen},
volume={7},
publisher={Akademie-Verlag},
place={Berlin},
},
date={1979},
pages={21--33},
review={\MR{569571 (81d:18004)}},
}

I also think that the key construction is explained somewhere in this book:

\bib{MR2178101}{article}{
author={Barr, Michael},
author={Wells, Charles},
title={Toposes, triples and theories},
note={Corrected reprint of the 1985 original [MR0771116]},
journal={Repr. Theory Appl. Categ.},
number={12},
date={2005},
pages={x+288},
review={\MR{2178101}},
}

However, I do not have my copy to hand so I cannot easily check this. The main idea
is that the problem can be formulated equationally. For each parallel pair
$f,g\:X\to Y$ we need an object $E(f,g)$ and a morphism $k(f,g)\:E(f,g)\to X$. There
should be a map $X\to E(1_X,1_X)$ that is inverse to $k(1_X,1_X)$ and also maps
$m(f,g,h)\:E(f,g)\to E(hf,hg)$ for all $h\:Y\to Z$. By some cunning means that I
don't remember one can write down a few equations for the operators $k(-,-)$ and
$m(-,-,-)$ that encapsulate the universal property of equalisers.