  An algebraic left Kan extension is a left Kan extension which interacts well
with the algebraic structure present in the given situation, and these appear
in various subjects such as the homotopy theory of operads and in the study of
conformal field theories. In the most interesting examples, the functor along
which we left Kan extend goes between categories that enjoy universal
properties which express the meaning of the calculation we are trying to
understand. These universal properties say that the categories in question are
universal examples of some categorical structure possessing some kind of
internal structure, and so fall within the theory of internal algebra
classifiers described in earlier work of the author. In this article conditions
of a monad-theoretic nature are identified which give rise to morphisms between
such universal objects, which satisfy the key condition of Guitart-exactness,
which guarantees the algebraicness of left Kan extending along them. The
resulting setting explains the algebraicness of the left Kan extensions arising
in operad theory, for instance from the theory of Feynman categories of
Kaufmann and Ward, generalisations thereof, and also includes the situations
considered by Batanin and Berger in their work on the homotopy theory of
algebras of polynomial monads.
