  This paper is a sequel to arXiv:2307.13358 and arXiv:2308.16090. A
construction associating a semialgebra with an algebra, subalgebra, and a
coalgebra dual to the subalgebra played a central role in the author's book
arXiv:0708.3398. In this paper, we extend this construction to certain
nonunital algebras. The resulting semialgebra is still semiunital over a
counital coalgebra. In particular, we associate semialgebras to locally finite
subcategories in $k$-linear categories. Examples include the Temperley-Lieb and
Brauer diagram categories and the Reedy category of simplices in a simplicial
set.
