Lambek's Lemma

Let $F$ be an endofunctor in $\mathcal{C}$. If $(X,\alpha )$ is an initial $F$-algebra then $\alpha$ is an isomorphism in $\mathcal{C}$. Thus, every initial algebra is a fixed point.

Specifically $\alpha :FX\cong X$

The dual holds for terminal coalgebra.

References

https://ncatlab.org/nlab/show/initial+algebra+of+an+endofunctor https://people.cs.nott.ac.uk/psztxa/mgs.2025/