Some time ago, in connection with trying to understand a construction of Amitsur (Embeddings of matrix rings, [Pac JM 36 (1971)](http://projecteuclid.org/euclid.pjm/1102971264), I stumbled across a short paper which considered some variant of the following question:



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> given (forgetful) functors $H: {\mathcal C} \to {\mathcal B}$ and $G:{\mathcal B}\to {\mathcal A}$, such that $GH:{\mathcal C}\to {\mathcal A}$ and $G:{\mathcal B}\to {\mathcal A}$ both have left adjoints, when can we deduce that $H$ has a left adjoint?
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I think in the intended application ${\mathcal A}$ was the category of sets and ${\mathcal B}$ was the category of rings; the idea being that it is easier or more transparent to construct "the free widget on a generating set" than "the free widget on a generating ring".


Unfortunately, I can't remember the name of the author nor the title of the paper. Does this sound familiar to anyone? and if not, does anyone have an alternative reference? Sorry to ask such a vague question, but I have been racking my brains and searching on MathSciNet to no avail.


UPDATE: while reading Todd Trimble's answer below, the veil inexplicably lifted and I was able to remember the author's surname, from which a quick Google turned up what I was after:



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> [An Interpolation Theorem for Adjoint Functors](http://www.jstor.org/pss/2036771)
>  S. A. Huq, Proceedings of the American Mathematical Society
>  Vol. 25, No. 4 (Aug., 1970), pp. 880-883 
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Sorry to waste people's time!



Reference for factorization of left adjoints?
This should probably be a comment but I don't know how to include links in comments. 


Results of this type are often called *adjoint triangle theorems*. There are many such: see John Power's paper [A unified approach to the lifting of adjoints](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1988__29_1/CTGDC_1988__29_1_67_0/CTGDC_1988__29_1_67_0.pdf) for a summary (and a unified approach). 


A sufficient condition for $H$ to admit a left adjoint is that $C$ has coequalizers and that $G$ is ``of descent type''. (This means that if we form the monad $S$ on $A$ induced by $G$ and its left adjoint, then the induced functor $B\to A^S$ is fully faithful.) This includes the important examples of the type considered in Todd's answer. 


for a summary. 


