I would like to know if the cotangent complex (say of rings) commutes with coequalisers. More precisely, let $B_1\rightrightarrows B_2\rightarrow C$ be a coequaliser of $A$-algebras. Is then the cotangent complex $L_{C/A}$ the coequaliser of $L_{B_1/A}\rightrightarrows L_{B_2/A}$?

I suspect the answer is no but could not find a concrete example for this.

Thanks!

homotopycoequalizers, and homotopy colimits in general. Therefore, a good starting point to look for a counterexample is a coequalizer which is not a homotopy coequalizer. – Fernando Muro Jul 30 '13 at 8:41