The definition I am working with is the following, a category $\mathcal{C}$ with all small colmits is called locally presentable if
- it has a set of small objects $S\subset Obj(\mathcal{C})$
- every other object is a small colimit of elements of $S$
Note: When I say category I mean a locally small category i.e. all Homs are proper sets. Here is how far I got, let $X\in Obj(\mathcal{C})$ we want to show that there is a cardinal $\kappa$ such that for every $\kappa$-filtered functor $F:I\to \mathcal{C}$ , we have $$Hom(X,colim_i F(i))\cong colim_i Hom(X, F(i))$$. Now $X=colim_j G(j)$ where $G:J\to \mathcal{C}$ and $G(j)\in S\ \forall i$ Therefore, by property of Hom, $$Hom(X,colim_i F(i))\cong lim_jHom(G(j), colim_i F(i))$$, and as each G(j) is small (we choose $\kappa$ which holds for all elements in S) $$lim_jHom(G(j), colim_i F(i))\cong lim_j colim_iHom(G(j), F(i))$$ Now for the required isomorphism, I need the limit and colimit to commute and I don't know how to do that.