$C$ is equivalent to a finitary algebraic category in the sense of Example 5.14(a). (Recall that these categories are those whose objects are sets with finitary operations satisfying some equations, and morphisms are homomorphisms between them.)

$C$ is equivalent to the category of $Set$-models of a Lawvere theory.

$C\simeq {Set}^{𝕋}$ for a finitary monad $𝕋$ on $Set$.