Steve Lack writes: You ask about a sketch for cartesian closed categories. Have a look at
at the paper "A presentation of topoi as algebraic relative to categories or graphs (Dubuc-Kelly, J. Alg. 81: 420-433, 1983). This describes something even tighter: the category of cartesian closed categories is monadic over the category of graphs.
Thanks! And thanks to everyone else for their helpful comments. I'm behind on answering my emails. In this approach, does each pair of objects in a ccc come with a chosen product and exponential? Are the morphisms of ccc's are required to preserve these on the nose? At first I was a bit shocked to hear of a sketch for ccc's, because the internal hom is contravariant in one variable. But I guess that as long as we treat ccc's purely 1-categorically that's no problem. But then I guess we pay the price of "undue strictness". Right? Best, jb