Profunctor Algebra

Let $P:\mathcal{D}^{\mathrm{op}}\times\mathcal{D}\to\mathrm{Sets}$ be a Profunctor. A $P$-algebra consists of a functor $F:\mathcal{C}\to\mathcal{D}$ and an Extranatural Transformation $\alpha:\star\to P(F(-),F(=))$.

If $\mathcal{C}$ is the Terminal Category, then this reduces to a section $\alpha:P(X,X)$. Moreover, if $P=\mathcal{D}(F(-),G(=))$ for some Functors $F,G$, then this further specializes to the definition of a Functor Dialgebra.