On colimits and elementary embeddings

We give a sharper version of a theorem of Rosicky, Trnkov ´ a and Ad ´ amek [13], and a ´ new proof of a theorem of Rosicky [12], both about colimits in categories of structures. Unl ´ ike the original proofs, which use category-theoretic methods, weuse set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C (n) -extendible cardinals.