If I am not mistaken, every small category and every algebraic category can be fully embedded into Str. This seems to follow from old results of the Czech school of category theory, see e.g. J. Algebra 11(1969), 159-212 and references listed there. If my memory does not deceive me, the same is true for every concrete category, provided one accepts Vopenka's Principle. From: Jim Lambek <lambek@triples.math.mcgill.ca> +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++