# Non-counital coalgebras

For any unital algebra $$A$$, we have an associated dual coalgebra $$A^{\circ}$$. (Recall that it is defined to be the largest subalgebra of the $$\mathbf{C}$$-linear dual of $$A$$ such that the coproduct $$\Delta(f)(a,b) = f(ab)$$ is well-defined.) What is the corresponding construction for a non-unital algebra. The coproduct part still works, but we have no counit, since this should arise as the dual of the unit. So are non-counital coalgebras studied in the literature? If so what are some references?