Lawvere theory

In category theory, a Lawvere theory (named after American mathematician William Lawvere) is a category that can be considered a categorical counterpart of the notion of an equational theory.

be a skeleton of the category FinSet of finite sets and functions.

Formally, a Lawvere theory consists of a small category L with (strictly associative) finite products and a strict identity-on-objects functor

A map between Lawvere theories (L, I) and (L′, I′) is a finite-product preserving functor that commutes with I and I′.

Lawvere theories together with maps between them form the category Law.