[hackandthink]

Lawvere theories should be fine as well:

"The rough idea is to define an algebraic theory as a category with finite products and possessing a “generic algebra” (e.g., a generic group), and then define a model of that theory (e.g., a group) as a product-preserving functor out of that category."

https://ncatlab.org/nlab/show/Lawvere+theory#the_theory_of_g...