Models of Enhanced 2-sketches & Algebras over Enhanced 2-monads
We study the enhanced 2-category of models of enhanced limit 2-sketches with tight weighted cones. We show that for any enhanced limit 2-sketch \mathbb{T} with tight cones, the enhanced 2-category \mathbb{M}\mathrm{od}_{s, w}(\mathbb{T}, \mathbb{K}) of models of \mathbb{T} in a locally presentable enhanced 2-category \mathbb{K}, in which the tight and the loose morphisms are the \mathscr{F}-natural transformations and the loose w-natural transformations, respectively, is equivalent to the enhanced 2-category {\mathrm{T}\text{-}\mathbb{A}\mathrm{lg}}_{s, w} of algebras over an enhanced 2-monad T on the models \mathbb{M}\mathrm{od}(\mathcal{T}_\tau, \mathbb{K}) restricted to the tight morphisms in \mathbb{T} with strict T-morphisms and w-T-morphisms.
Along the way, we establish an enriched analogue of the Orthogonal Sub-category Theorem, and generalise results on the reflectivity and the monadicity of models of enriched limit sketches in the base of enrichment to any arbitrary locally presentable enriched category.