ct.category theory - Why forgetful functors usually have LEFT adjoint?

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

• $C$ is a presentable category, 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 monoids, 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 $[1]$. (One actually needs only the full subcategory of $Delta^mathrm{op}$ on the objects $[0]$, $[1]$, $[2]$, $[3]$, and the cones involving these objects; expanding this gives a possibly more familiar presentation of the notion of monoid.)

In these cases (which include models of any essentially algebraic theory) 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, Locally presentable and accessible categories, for an excellent introduction to the subject.

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