  The theory of presentations of enriched monads was developed by Kelly, Power,
and Lack, following classic work of Lawvere, and has been generalized to apply
to subcategories of arities in recent work of Bourke-Garner and the authors. We
argue that, while theoretically elegant and structurally fundamental, such
presentations of enriched monads can be inconvenient to construct directly in
practice, as they do not directly match the definitional procedures used in
constructing many categories of enriched algebraic structures via operations
and equations.
  Retaining the above approach to presentations as a key technical
underpinning, we establish a flexible formalism for directly describing
enriched algebraic structure borne by an object of a $V$-category $C$ in terms
of parametrized $J$-ary operations and diagrammatic equations for a suitable
subcategory of arities $J \hookrightarrow C$. On this basis we introduce the
notions of diagrammatic $J$-presentation and $J$-ary variety, and we show that
the category of $J$-ary varieties is dually equivalent to the category of
$J$-ary $V$-monads. We establish several examples of diagrammatic
$J$-presentations and $J$-ary varieties relevant in both mathematics and
theoretical computer science, and we define the sum and tensor product of
diagrammatic $J$-presentations. We show that both $J$-relative monads and
$J$-pretheories give rise to diagrammatic $J$-presentations that directly
describe their algebras. Using diagrammatic $J$-presentations as a method of
proof, we generalize the pretheories-monads adjunction of Bourke and Garner
beyond the locally presentable setting. Lastly, we generalize Birkhoff's Galois
connection between classes of algebras and sets of equations to the above
setting.
