On colimits and elementary embeddings

We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a new proof of a theorem of Rosicky, both about colimit preservation between categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as alpha-strongly compact and C^(n)-extendible cardinals.