I found the following statements in some lecture notes in category theory:
A functor between sets $S$ and $T$ is a function from $S$ to $T$
A functor between two groups $G$ and $H$ is a group homomorphism from $G$ to $H$
A functor between two posets $P$ and $Q$ is a monotone function from $P$ to $Q$
Consider the second statement. A group was defined as a category with one object in which every arrow is an isomorphism. Then what is the definition of a group homomorphism in this setting? What exactly do I need to check to establish the claim? Here is what I have. Suppose $G$ is the category with object $\ast$ and $H$ is a category with object $\clubsuit$ and let $\alpha:G\to H$ be a functor. By the definition of a functor, $\alpha(\ast)=\clubsuit$, and if $g:\ast\to \ast$ is an arrow in $G$, then $\alpha(g):\clubsuit\to \clubsuit$ is an arrow in $H$. The remaining conditions on $\alpha $ are: $\alpha(g_1\circ g_2)=\alpha(g_1)\circ \alpha(g_2)$ and $\alpha(id_\ast)=id_\clubsuit$. But without knowing the definition of a group homomorphism when a group is defined as a category, I don't know how to proceed.
A poset can also be considered as a skeletal preorder. (A preorder is a category in which there is at most one arrow between any two objects.) In this definition, what is the definition of a monotone map between such categories?
A set is a category where the objects are the elements of a set and the only arrows are the identity arrows. What is a function between such categories?