F-coalgebra

Remarks

The $F$-coalgebras and their homomorphisms form a category. An $F$-coalgebra is the same as an algebra for the functor $F^{\mathrm{op}}$ on $\mathcal{C}^{\mathrm{op}}$.

Example: on the category of sets let $F(X) = \N \times X$, so $F(f) = \id_{\N} \times f$. The map $\alpha \colon \N \to \N \times \N$ with $\alpha(n) = (n, n + 1)$ makes $(\N, \alpha)$ an $F$-coalgebra. The set $\N^{\N}$ of streams with $\sigma(s) = (s_0, (s_1, s_2, \dots))$ is the terminal $F$-coalgebra. The unique homomorphism from $(\N, \alpha)$ sends $n$ to the stream $(n, n + 1, n + 2, \dots)$.

Non-example: the map $f \colon \N \to \N$ with $f(n) = n + 1$ is not a homomorphism from $(\N, \alpha)$ to itself. Indeed $\alpha(f(n)) = (n + 1, n + 2)$, while $F(f)(\alpha(n)) = (n, n + 2)$.