Suppose we have two categories $D$ and $S$, as well as two functors $K,L:D\to S$ and a natural transformation $\varphi:K\to L$.
Given another category $C$ and a functor $Y:C\to S^D$, is there a nice way to denote the functor from $(Y\downarrow K)$ to $(Y\downarrow L)$ induced by $\varphi$?
I'll expose the motivation behind this question: Suppose that $K,L:\mathbf\Delta^{op}\to\text{Set}$ are simplicial sets, and $\varphi:K\to L$ is a morphism in the category of simplicial sets. I think we can define the simplex category of $K$ as the comma category $$\Delta\downarrow K$$ where $\Delta:\mathbf\Delta\rightarrow\text{sSet}$ is the Yoneda embedding for $\mathbf\Delta$ defined by $\phi\mapsto\hom_{\mathbf\Delta}(-,\phi)$ for an object or morphism $\phi$ of $\mathbf\Delta$.
I'd like to find an elegant way to denote the functor from $(\Delta\downarrow K)$ to $(\Delta\downarrow L)$ induced by $\varphi$. However, I'm not sure if the notation $(\Delta\downarrow\varphi):(\Delta\downarrow K)\to(\Delta\downarrow L)$ makes any formal sense!
Any ideas?