Many standard examples of algebraic "forgetful" functors $U : C \to \mathrm{Set}$ have the following form:

* $C$ is a <i>presentable category</i>, i.e., there is a small category $I$ and a collection $S$ of cones of $I$ such that $C$ is equivalent to the full subcategory of functors $I \to \mathrm{Set}$ consisting of those functors which send the cones of $S$ to limit diagrams in $\mathrm{Set}$;
* $U$ is evaluation at an object $u \in I$.

For example, if $C$ is the category of groups, take $I = \Delta^{\mathrm{op}}$ so that functors $I \to \mathrm{Set}$ are simplicial sets and choose $S$ so that the objects of $C$ are those simplicial sets $X$ such that $X_0 = \ast$ and $X_{i+j} \to X_{i} \times X_{j}$ is an isomorphism (where this map is induced by the inclusions of the first $i+1$ and last $j+1$ elements of an ordered $i+j+1$ element set).  The object $u$ is the two-element set $[1]$.

In these cases (which include models of any <a href="http://ncatlab.org/nlab/show/essentially+algebraic+theory">essentially algebraic theory</a>) the existence of a left adjoint is guaranteed by the theory of presentable categories.  Indeed, the inclusion of $C$ into $\mathrm{Set}^I$ has a left adjoint which we compose with the constant diagram functor $\mathrm{Set} \to \mathrm{Set}^I$ to obtain a left adjoint to $U$.  See Adamek and Rosicky, <i>Locally presentable and accessible categories</i>, for an excellent introduction to the subject.