| context | $F$ in ${\bf C}\longrightarrow{\bf C}$ |
| definiendum | $\mathcal{A}:\mathrm{Ob}_\mathrm{it}$ |
| postulate | $\mathcal{A}$ … $F$-algebra |
| definiendum | $\langle f\rangle:\mathrm{it}[\langle A,\alpha\rangle, \langle B,\beta\rangle]$ |
| postulate | $f\circ\alpha=\beta\circ F(f)$ |
The category of F-algebras and F-algebra homomorphisms. The postulate says that it can't matter if you perform the operation ($\alpha$ resp. $\beta$) before or after the transformation $f$.
Note that $\alpha,\beta,f$ are arrows in ${\bf C}$, while $\langle f\rangle$ denotes the arrow between $F$-algebras $\langle A,\alpha\rangle$ and $\langle B,\beta\rangle$ corresponding to the homomorphism $f$. Clearly, $\langle f\rangle$ and $f$ are in bijection and one often just writes $f$ for both.
Wikipedia: F-algbera