Dear Michael, Your first question
Does anyone know whether every limit closed full subcategory of an equational category is the category of models of a limit sketch?
depends on set theory: if Vopenka's Principle (VP) holds, the answer is affirmative, see Corrollary 6.24 in "Locally Presentable and Accessible Categories". If VP does not hold, the answer is negative. Our counterexample 6.25 works the category Rel \Sigma which is not equational, but which can embedded as a full, limit-closed subcategory of an equational category (see e.g. Theorem 1.46 in the book).
Assuming that is true and the equational category is finitary, if the subcategory is also closed in the equational category under filtered colimits, is also the category of models an FL-sketch?
The answer is affirmative (absolutely): a full subcategory of a locally finitely presentable category closed under limits and filtered colimits is locally finitely presentable, see Corollary 2.48 in the above book. Best regards Jiri [For admin and other information see: http://www.mta.ca/~cat-dist/ ]