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Here are some basic remarks and examples: (Caution. This answer refers to preorders; but many of the remarks also apply to partially ordered sets aka posets)

• Many concepts of category theory have a nice illustration when applied to preorders; but also the other way round: Many concepts familiar from preorders carry over to categories (for example suprema motivate colimits; see also below).

• This is partially justified by the following observation: An arbitrary category is a sort of a preorder but where you have to specify in addition a reason why $x \leq y$, in form of an arrow $x \to y$. The axioms for a category tell you: For every $x$ there is a distinguished reason for $x \leq x$, and whenever you have a reason for $x \leq y$ and for $y \leq z$, you also get a reason for $x \leq z$.

• A preorder is a category such that every diagram commutes.

• In a preorder, the limit of a diagram is the same as the infimum of the involved objects. Similarly, a colimit is just a supremum. The transition morphisms don't matter.

• When $f^* : P \to Q$ is a cocontinuous functor between preorders, where $P$ is complete, then $f^*$ has a right adjoint $f_*$; you can write it down explicitly: $f_*(q)$ is the infimum of the $p$ with $f^*(p) \leq q$. This construction motivates the General Adjoint Functor Theorem. In this setting we only have to add the solution set condition, so that the a priori big limit can be replaced by a small one and therefore exists.

• Let $f : X \to Y$ be a map of sets. Then the preimage functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ between the power sets is right adjoint to image functor $\mathcal{P}(X) \to \mathcal{P}(Y)$. Every cocontinuous monoidal functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ arises this way.

• The inclusion functor $\mathrm{Pre} \to \mathrm{Cat}$ has a left adjoint: It sends every category to its set of objects with the order $x \leq y$ if there is a morphism $x \to y$. In particular, it preserves all limits. In fact, it creates all limits, and limits in $\mathrm{Cat}$ are constructed "pointwise". Thus, the same is true for limits in $\mathrm{Pre}$ (which one could equally well see directly). For example, the pullback of $f : P \to Q$ and $g : P' \to Q$ is the pullback of sets $P \times_Q P'$ equipped with the order $(a,b) \leq (c,d)$ iff $a \leq c$ and $b \leq d$. If we apply this to difference kernels, we see that $f : P \to Q$ is a monomorphism iff the underlying map of $f$ is injective.

• The forgetful functor $\mathrm{Pre} \to \mathrm{Set}$ creates coproducts: Take the disjoint union $\coprod_i P_i$ and take the order $a \leq b$ iff $a,b$ lie in the same $P_i$, and with respect to that preorder we have $a \leq_i b$.

• The construction of coequalizers seems to be more delicate; see this SE discussion.

I don't have a reference for all these observations, but they are easy. A general reference for basic category-theoretic constructions (and it surely says something about preorders and posets) is the book "Abstract and Concrete Categories - The Joy of Cats" by Adamek, Herrlich, Strecker which you can find online.

EDIT: Here is something not so basic: Sefi Ladkani studied the notion of derived equivalent posets. Two posets $X,Y$ are called (universally) derived equivalent if for some specific (every) abelian category $\mathcal{A}$ the diagram categories $\mathcal{A}^X$, $\mathcal{A}^Y$ are derived equivalent.

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Here are some basic remarks and examples: (Caution. This answer refers to preorders; but many of the remarks also apply to partially ordered sets aka posets)

• Many concepts of category theory have a nice illustration when applied to posetspreorders; but also the other way round: Many concepts familiar from posets preorders carry over to categories (for example suprema motivate colimits; see also below).

• This is partially justified by the following observation: An arbitrary category is a sort of a poset preorder but where you have to specify in addition a reason why $x \leq y$, in form of an arrow $x \to y$. The axioms for a category tell you: For every $x$ there is a distinguished reason for $x \leq x$, and whenever you have a reason for $x \leq y$ and for $y \leq z$, you also get a reason for $x \leq z$.

• A poset preorder is a category such that every diagram commutes.

• In a posetpreorder, the limit of a diagram is the same as the infimum of the involved objects. Similarly, a colimit is just a supremum. The transition morphisms don't matter.

• When $f^* : P \to Q$ is a cocontinuous functor between posetspreorders, where $P$ is complete, then $f^*$ has a right adjoint $f_*$; you can write it down explicitly: $f_*(q)$ is the infimum of the $p$ with $f^*(p) \leq q$. This construction motivates the General Adjoint Functor Theorem. In this setting we only have to add the solution set condition, so that the a priori big limit can be replaced by a small one and therefore exists.

• Let $f : X \to Y$ be a map of sets. Then the preimage functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ between the power sets is right adjoint to image functor $\mathcal{P}(X) \to \mathcal{P}(Y)$. Every cocontinuous monoidal functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ arises this way.

• The inclusion functor $\mathrm{Pos} \mathrm{Pre} \to \mathrm{Cat}$ has a left adjoint: It sends every category to its set of objects with the order $x \leq y$ if there is a morphism $x \to y$. In particular, it preserves all limits. In fact, it creates all limits, and limits in $\mathrm{Cat}$ are constructed "pointwise". Thus, the same is true for limits in $\mathrm{Pos}$ \mathrm{Pre}$(which one could equally well see directly). For example, the pullback of$f : P \to Q$and$g : P' \to Q$is the pullback of sets$P \times_Q P'$equipped with the order$(a,b) \leq (c,d)$iff$a \leq c$and$b \leq d$. If we apply this to difference kernels, we see that$f : P \to Q$is a monomorphism iff the underlying map of$f$is injective. • The forgetful functor$\mathcal{Pos} \mathrm{Pre} \to \mathrm{Set}$creates coproducts: Take the disjoint union$\coprod_i P_i$and take the order$a \leq b$iff$a,b$lie in the same$P_i$, and with respect to that poset preorder we have$a \leq_i b$. • The construction of coequalizers seems to be more delicate; see this SE discussion. I don't have a reference for all these observations, but they are easy. A general reference for basic category-theoretic constructions (and it surely says something about preorders and posets) is the book "Abstract and Concrete Categories - The Joy of Cats" by Adamek, Herrlich, Strecker which you can find online. 1 [made Community Wiki] Here are some basic remarks and examples: • Many concepts of category theory have a nice illustration when applied to posets; but also the other way round: Many concepts familiar from posets carry over to categories (for example suprema motivate colimits; see also below). • This is partially justified by the following observation: An arbitrary category is a sort of a poset but where you have to specify in addition a reason why$x \leq y$, in form of an arrow$x \to y$. The axioms for a category tell you: For every$x$there is a distinguished reason for$x \leq x$, and whenever you have a reason for$x \leq y$and for$y \leq z$, you also get a reason for$x \leq z$. • A poset is a category such that every diagram commutes. • In a poset, the limit of a diagram is the same as the infimum of the involved objects. Similarly, a colimit is just a supremum. The transition morphisms don't matter. • When$f^* : P \to Q$is a cocontinuous functor between posets, where$P$is complete, then$f^*$has a right adjoint$f_*$; you can write it down explicitly:$f_*(q)$is the infimum of the$p$with$f^*(p) \leq q$. This construction motivates the General Adjoint Functor Theorem. In this setting we only have to add the solution set condition, so that the a priori big limit can be replaced by a small one and therefore exists. • Let$f : X \to Y$be a map of sets. Then the preimage functor$\mathcal{P}(Y) \to \mathcal{P}(X)$between the power sets is right adjoint to image functor$\mathcal{P}(X) \to \mathcal{P}(Y)$. Every cocontinuous monoidal functor$\mathcal{P}(Y) \to \mathcal{P}(X)$arises this way. • The inclusion functor$\mathrm{Pos} \to \mathrm{Cat}$has a left adjoint: It sends every category to its set of objects with the order$x \leq y$if there is a morphism$x \to y$. In particular, it preserves all limits. In fact, it creates all limits, and limits in$\mathrm{Cat}$are constructed "pointwise". Thus, the same is true for limits in$\mathrm{Pos}$(which one could equally well see directly). For example, the pullback of$f : P \to Q$and$g : P' \to Q$is the pullback of sets$P \times_Q P'$equipped with the order$(a,b) \leq (c,d)$iff$a \leq c$and$b \leq d$. If we apply this to difference kernels, we see that$f : P \to Q$is a monomorphism iff the underlying map of$f$is injective. • The forgetful functor$\mathcal{Pos} \to \mathrm{Set}$creates coproducts: Take the disjoint union$\coprod_i P_i$and take the order$a \leq b$iff$a,b$lie in the same$P_i$, and with respect to that poset we have$a \leq_i b\$.

• The construction of coequalizers seems to be more delicate; see this SE discussion.