A variety is called Jónsson-Tarski if its algebraic theory contains a unique constant and a binary operation \( + \) satisfying \( x + 0 = 0 + x = x \).
It is well known that this is equivalent to requiring the pair of morphisms \( \langle 1_X , 0\rangle\colon X \to X \times Y \) and \( \langle 0, 1_Y\rangle\colon Y \to X \times Y \) to be jointly strongly epimorphic. Since this pair of morphisms can be defined in any pointed category with finite limits, this motivated the definition of a unital category.

There did not exist a categorical version of operations of theories of varieties such as unital, strongly unital, subtractive and Mal'tsev varieties. Bourn and Janelidze introduced the notion of approximate Mal'tsev operation as a categorical version of the varietal Mal'tsev operation in [1].

The aim of this talk is to do a similar approach to the unital context. The approximate addition we define in the context of unital category is a tool that can be used to prove several results in this context, following a similar proof as in the varietal case. Thus, the talk aims to present this categorical approach to unitality and to investigate the strength of approximate addition in the categorical setting.

References
[1] D. Bourn, Z. Janelidze Approximate Mal'tsev Operation, Theory and Applications of Categories, Vol. 21, No. 8, 2008, pp. 152-171.