When reading up the general theory of cofibrantly generated model categories, anything related to the small object argument relies on the choice of some cardinal $\lambda$. I.e. we need $\lambda$-sequences to define $\lambda$-smallness, and a transfinite composition of a $\lambda$-sequence of pushouts of coprducts when applying the small object argument.
However, when Thomason uses the small object argument in order to prove that $\mathbf{Cat}$ (with generating cofibrations and generating trivial cofibrations as chosen) is indeed a cofibrantly generated model category, he doesn't even mention any cardinals and just talks about ordinary sequences.
My question is why. My guts tell me, that due to $\mathbf{Cat}$ being locally finitely presentable, he may chose $\lambda = \aleph_0$. I am however not familiar enough with the theory of locally finitely presentable categories to find a convincing argument why.
Moreover, does this imply that any $\lambda$-filtered colimit in $\mathbf{Cat}$ (or any other locally finitely presentable category) is in fact $\aleph_0$-filtered?