I am studying what coinduction is. In particular, I am reading that coinductive datatypes can be defined as elements of a final coalgebra for a given polynomial endofunctor on $\tt Set$. I've seen that $A^w$, infinite streams over an alphabet $A$, is the final coalgebra of the functor $FX \rightarrow A \times X$, and similarly $A^\infty$ (finite and infinity streams over $A$) is the is the final coalgebra of $FX = {\tt id} + A × X$.
I'm wondering if the set of finite streams over $A$ is a final coalgebra for some polynomial endofunctor on $\tt Set$. In other words, if there is a way to apply coinduction over the set of finite streams over $A$. I've seen that there must exist a so called $\textit{structure map}$ that maps $X \rightarrow A \times X$ and such that for $X=$our coalgebra, the previous map is an isomorphism (Lambek’s lemma).
Our object of study is $\bigoplus A$ and clearly $A \times \bigoplus A$ is isomorphic to it, so in principle $\bigoplus$ could be a coalgebra (Lambek’s lemma is currently my only source for providing counterexamples).
I find the notion of coinduction a bit counterintuitive because I don't know yet why it is well founded, but I think it would be a bit enlightening to know if you can use coinduction over streams of finite length.
