Posetalization Functor

Let $(P,⪯)$$(P,\preceq)$ be a pre-order set, $\pi (a)=\pi (b)⟺a⪯b\wedge b⪯a$$\pi(a)=\pi(b)\iff a\preceq b\wedge b\preceq a$.

Then $\pi (a)\le b⟺a⪯{\pi }^{-1}(b)$$\pi(a)\le b\iff a\preceq \pi^{-1}(b)$, hence it is an adjoint functor.

Moreover, let $F$$F$ be the posetalization functor and i be the inclusion functor. Then we have the following adjoint.

$$\begin{array}{}\text{(1)}& {\mathrm{Hom}}_{\mathrm{Pos}}(F(P),Q)\cong {\mathrm{Hom}}_{\mathrm{Pre}}(P,i(Q))\end{array}$$

The proof follows from the commutative diagram directly. Since $Q$$Q$ is a partial order set, so any pre-order homomorphism must factor through $\pi$$\pi$.

i.e. $a\le b\wedge b\le a\Rightarrow f(a)\le f(b)\wedge f(b)\le f(a)\Rightarrow f(a)=f(b)$$a\le b\wedge b\le a\Rightarrow f(a)\le f(b)\wedge f(b)\le f(a)\Rightarrow f(a)=f(b)$