Does the cotangent complex commute with coequalisers?

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!

-
I guess, as you do, that the answer is not, but the cotangent complex is the total left derived functor of a left Quillen functor, hence it does commute with homotopy coequalizers, 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
Fernando, thanks for the comment. Where could I find an introduction to homotopy coequalisers, respectively some examples which could be tested explicitly? Thanks! –  user36504 Aug 7 '13 at 14:31
Hi user, you're welcome. I'm afraid I don't know of any gentle introduction. You would have to read about homotopy colimits in general. That you can do in any book on model categories, e.g. Hovey's. Depending on your backgrouund on homotopy theory, it might be wiser to catch a homotopy theorist, make her/him sit down with you and translate the general notions to the context you're interested in. –  Fernando Muro Aug 8 '13 at 8:14