TITLE: A finite object in Set is a finite Set
QUESTION [5 upvotes]: I'd like to prove that an object of $\bf{Set}$ is finite iff it is a finite set. Finite object is defined as being small with respect to a finite cardinal; here's the definition of smallness I'm assuming:
"Let $k$ be a cardinal and $\mathscr{C}$ a cocomplete category. An object $A \in \mathscr{C}$ is said to be $\textit{$k$-small}$ iff for all $k$-filtered ordinal $\lambda$ (which means that $cof(\lambda) >k$ strictly) and for all  colimit preserving-functors $X: \lambda \to \mathscr{C}$ we have that the canonical morphism of sets $$colim_{\beta<\lambda}\mathscr{C}(A,X_{\beta}) \to \mathscr{C}(A,colim_{\beta < \lambda} X_{\beta})$$ is an isomorphism".
I'm able to prove that "finite set$\implies$finite object", but the converse gives me some problems. I tried to suppose that it's an infinite set and to use it's cardinality as an infinite cardinal that gives an absurd with such an iso, but I got stuck..
Any help or any hint is welcome! Thanks in advance :)

REPLY [4 votes]: I'm missing something or you only lack the easy side of [LPAC] Example 1.2.1?
Every set is the directed colimit of its finite subsets. Hence if $K$ is a finite object, then you must have that $1_K\colon K\to K=\varinjlim{}_{F\in \mathcal P_0(K)} F$ must factor through one of the inclusions of a finite subset of $K$; hence $K$ is a finite set.