Fix an endofunctor $𝐹:\mathrm{𝒞︀}\to \mathrm{𝒞︀}$. The $𝐹$-algebras form a displayed category $\text{AlgStr}(𝐹)$ over $\mathrm{𝒞︀}$.

Over an object $𝑥$, a displayed object of $\text{AlgStr}(𝐹)$ is a structure map

Over $𝑓:𝑥\to 𝑦$, a displayed morphism from $𝛼$ to $𝛽$ is the proposition that $𝑓$ is an algebra homomorphism:

The total category $\text{Alg}(𝐹)=\int \text{AlgStr}(𝐹)$ is the category of $𝐹$-algebras.