Proposition [efr-9O7K]

Let \mathcal {T},\mathcal {T}' be limit sketches and suppose given a functor F: \mathsf {Mod}(\mathcal {T}) \to \mathsf {Mod}(\mathcal {T}') which preserves limits and is accessible, that is it preserved \kappa -filtered colimits for some \kappa . Then F admits a left adjoint L.

In particular, F(A)(S) = \mathsf {Mod}(\mathcal {T})(L(y(S)),A)