Lincs completion–final INC coalgebra comparison conjecture

Let $\mathsf{T}_{\mathsf{INC}}$ denote the tangent endofunctor on $\mathsf{INC}(\mathcal{C})$, and suppose either preceding existence theorem applies, so that the universal Lincs completion and the resulting final INC coalgebra exist. Here $\nu\mathsf{T}_{\mathsf{INC}}$ denotes the final coalgebra of $\mathsf{T}_{\mathsf{INC}}$. Lincs completion–final INC coalgebra comparison conjecture. Under the hypotheses of either preceding existence theorem, the universal Lincs completion is equivalent to the resulting final INC coalgebra:

$$\mathsf{LINCS}(\mathcal{C})\simeq\nu\mathsf{T}_{\mathsf{INC}},$$

with the equivalence compatible with the embedding of base learning models and their iterated tangent factorization problems. This compares the proposed universal completion with the coalgebraic object supplied by the preceding existence theorems; the source gives no resolution of the comparison.