Let $\mathcal{C}$, $\mathcal{C'}$, $\mathcal{D}$ and $\mathcal{E}$ be four categories. Let $F$, $G$ and $K$ be three functors as follows: $\mathcal{C'} \xrightarrow{K} \mathcal{C} \xrightarrow{F} \mathcal{E} \xleftarrow{G} D$. We have a functor from the category $F \circ K \downarrow G$ to the category $F \downarrow G$ that
- takes any triple $(c', d, \alpha)$, where $c'$ is an object of $\mathcal{C'}$, $d$ an object of $\mathcal{D}$ and $\alpha$ a morphism $F(K(c')) \to G(d)$ in $\mathcal{E}$, and returns the triple $(K(c'), d, \alpha)$;
- takes any morphism $(\beta, \gamma): (c'_1, d_1, \alpha_1) \to (c'_2, d_2, \alpha_2)$, where $\beta$ is a morphism $c'_1 \to c'_2$ in $\mathcal{C'}$ and $\gamma$ is a morphism $d_1 \to d_2$ in $\mathcal{D}$, and returns $(K(\beta), \gamma)$.
What is a reasonable (standard?) notation/name for this functor?