Definition. category of $F$-algebras [math-000G]

For a fixed endofunctor $F$ in a proper category $C$ (below notations omit fixed $F$),

1. the F-algebras $(x, \alpha)$ form the objects of the category,
2. morphisms are homomorphisms of objects $(x, \alpha) \xrightarrow{F\;m} (y, \beta)$, composition is given by functor laws.

A homomorphism is a $C$-morphism $m$ makes below diagram commutes.

We can extra check identity indeed works.