We say that an object $X$ of a category $C$ is $\kappa$-compact (also $\kappa$-presentable and $\kappa$-accessible) for a cardinal $\kappa$ if $h^X(\cdot):=Hom(X,\cdot)$ commutes with all $\kappa$-filtered colimits.

In Makkai-Pare, a different but equivalent definition is given, that $X$ is $\kappa$-compact if for any $\kappa$-filtered functor $F:I\to C$ (i.e. the category $I$ is $\kappa$-filtered), any morphism $X\to \varinjlim F$ factors as $X\to F(i) \to \varinjlim F$, and any two factorizations $X\to F(i)\to \varinjlim F$ and $X\to F(i')\to \varinjlim F$ admit a majorant factorization $X\to F(i'')\to \varinjlim F$, that is, a factorization $X\to F(i'')\to \varinjlim F$ along with a commutative diagram

$$\begin{matrix} X&\to&F(i)\\ \downarrow&&\downarrow\\ F(i')&\to&F(i'')\\ \end{matrix}$$

such that $$F(i)\to \varinjlim F=F(i)\to F(i'')\to \varinjlim F$$ and $$F(i')\to \varinjlim F=F(i')\to F(i'')\to \varinjlim F$$ are the natural maps.

Let's try to show that they are equivalent:

Proof. '$\Rightarrow$' The existence of a factorization follows from the fact that the factorization exists in the category of sets, since giving a map $X\to \varinjlim F$ is the same as giving a map $$*\to h^X\varinjlim_i F(i)\cong\varinjlim_i h^XF(i),$$ and since we can give the colimit as a quotient of the disjoint union, the point $*\to \varinjlim_i h^XF(i)$ maps through some $h^XF(i)$, which gives us the necessary commutative triangle upon inspection.

Questions:

In the definition of Makkai-Pare, it seems like we should need the existence of a majorant factorization for any family of factorizations indexed by a $\kappa$-small set. Why is this not the case?

How can we prove the existence of the majorant factorization in the $\Rightarrow$ direction?

How do we prove the entire other direction?