nLab
algebra for an endofunctor

For a category $C$ and endofunctor $F$, an algebra of $F$ is an object $X$ in $C$ and a map $\alpha :F(X)\to X$. ($X$ is called the carrier of the algebra)

A morphism between two algebras $(X,\alpha )$ and $(Y,\beta )$ of $F$ is a morphism $m:X\to Y$ in $C$ such that the following square commutes:

$$\begin{array}{ccc}F(X)& \stackrel{F(m)}{\to }& F(Y)\\ \alpha \downarrow & & \downarrow \beta \\ X& \stackrel{m}{\to }& Y\end{array}.$$\array{ F(X) & \stackrel{F(m)}{\rightarrow} & F(Y) \\ \alpha\downarrow && \downarrow \beta \\ X & \stackrel{m}{\rightarrow} & Y } \,.

Composition of such morphisms of algebras is given by composition of the underlying morphisms in $C$.

The dual concept is a coalgebra for an endofunctor. Both algebras and coalgebras for endofunctors on $C$ are special cases of algebras for C-C bimodules.

If the endofunctor $F$ has the structure of a monad, then an algebra (aka module) over that monad is an algebra for $F$ that satisfies an additional associativity property. Morphisms between algebras for monads are simply morphisms between algebras for the underlying functors. For more details see algebra over a monad.

An initial object in the category of algebras for $F$ is an initial algebra of $F$.