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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.

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    While I don't know much about the technical details of coalgebras, the set of finite words of an alphabet is usually given by an algebra (i.e. inductive rules for generating longer strings from smaller strings). – 2017-02-16
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    I have doubts, for several reasons. The simplest being that the sub-coalgebras of terminal coalgebras that "track" arbitrary finite structures are still usually infinite themselves. (Think about the structure of the terminal coalgebra for $1+-$.) – 2017-03-22

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