How to complete semi-lattice via Adjunctions? I am thinking about free/complete lattice constructions from sets, posets and semilattices. For example, given a set $A$ we can construct (at least) two canonical posets from it as one being $(2^A,\subseteq)$ under inclusion order and the other being the trivial order (that is, $a\le b$ in $A$ iff $a=b$). I guess one can construct them as adjunctions to the forgetful functor $$\operatorname{Poset}\rightarrow\operatorname{Set}.$$ However I do not know how to obtain these left and adjoints. I have heard that one can write them as (co)limits, but never learn (or found a reference that explain in a way that I can understand) how to do it. So my first question is, can somebody explain me how to write explicit formulas for (possible) adjoints to this functor?
Next, I want to do the same thing for each forgetful functor in the chain
$$\operatorname{CompletelyDistributiveLattice}\rightarrow\operatorname{CompleteLattice}\rightarrow\operatorname{CompleteSemiLattice}\rightarrow\operatorname{SemiLattice}\rightarrow\operatorname{Poset}$$
Specially I was thinking about constructing complete semi-lattices from semi-lattices without hurting to existing (finite) joins. This is clearly related to the first question, since once I learn to obtain general left adjoints explicitly it become trivial.
 A: The left adjoint of the forgetful functor from posets to sets sends a set to the discrete poset on that underlying set, but there is no right adjoint; it's easy to check that if such a right adjoint $R$ existed then its value $R(S)$ on any set $S$ would have $s_1\le s_2$ and $s_2\le s_1$ for every $s_1,s_2$, so wouldn't be a poset at all.
For your problem of free complete semilattices, is a good idea to first notice that the left adjoint of the forgetful functor from complete semilattices to sets does send a set to its powerset, and a function to the associated direct image morphism of powersets. We can then readily solve the problem of constructing the free complete semilattice on a semilattice $S$, at least formally, by subjecting the powerset complete semilattice $\mathcal P(S)$ to the relations asserting that finitary joins in $\mathcal P(S)$ must be made to agree with their definition in $S$. That is, we quotient by the equivalence relation $\sim$ on $\mathcal P(S)$ that says a subset $\{a,b,\cdots\}$ is equivalent to the subset $\{a\vee b,\cdots\}$, with $\vee$ the supremum from $S$. Then complete semilattice maps out of $\mathcal P(S)/\sim$ are naturally identified with complete semilattice maps out of $\mathcal P(S)$ which respect $\sim$, or in other words which restrict to a semilattice homomorphism on $S$. Since complete semilattice maps out of $\mathcal P(S)$ in general are naturally identified with set functions out of $S$, this shows $\mathcal P(S)/\sim$ has the universal property of the free complete semilattice on $S$.
You can handle the free semilattice on a poset $T$ similarly: construct the free semilattice on $T$'s underlying set, namely the finite powerset of $T$, then quotient by the relations asserting that $\{a,b\}\sim b$ if $a\le b$ in $T$.
Interestingly, the forgetful functor from complete lattices to complete semilattices is a rare example of a functor between complete categories which preserves all limits but admits no left adjoint. The problem is that the free complete lattice on, say, the powerset of a three-element set $\{a,b,c\}$, forms a proper class. There is a construction of a proper class of distinct elements $p_\alpha$ given inductively by $p_{\alpha+1}=a\vee b\wedge c\vee a\wedge b\vee c\wedge p_\alpha$ (identifying $a$ with $\{a\}$ and associating, say, left to right) for successor ordinals and $p_\beta=\sup_{\alpha<\beta} p_\alpha$ for limit ordinals.
To summarize, sometimes showing a left adjoint exists is a bit easier than giving an explicit construction, but if it's an explicit construction you want then you need to rely on special aspects of your situation. There is a rather general answer for algebraic categories, like lattices, groups, rings, and complete lattices (defined by sets with total operations), which says that a left adjoint to a forgetful functor to sets takes $S$ to the object formed of all "words" constructed of elements of $S$ out of the operations of the algebraic category--at least if the class of such words is a set! And left adjoints between two such algebraic categories can be constructed out of the free construction on a set by imposing appropriate relations.
