Specific formulation of Yoneda lemma with comma category In a problem sheet there is a statement of the co-Yoneda lemma as follows (the whole question is included for context):



My question is, what would be the formulation of the usual Yoneda lemma in this setting? That is, using the diagonal and projection functors, and the category $(*\Rightarrow F)$.
I'm assuming that one must exist, given simply by 'reversing the arrows', but I can't find a statement of it anywhere.
 A: As usual, what you must realize is that $\mathcal{C}$ is entirely arbitrary in the context of the claim. Thus, we can replace it by $\mathcal{C}^{op}$. One then rewrites the text as follows:
Consider a functor $F: \mathcal{C} \to \text{Set}$. Define the category $* \Rightarrow F$ with objects consisting of pairs $(x, y)$ where $x \in \mathcal{C}$ and $y \in F(x)$ and morphisms given by morphisms $x_1 \to x_2$ inducing maps of pointed sets $F(x_1) \to F(x_2)$ (so this function sends $y_1$ to $y_2$).
There is a diagonal functor $\Delta: \mathcal{C}^{op} \to \text{Fun}((* \Rightarrow F)^{op}, \mathcal{C}) = \text{Fun}((* \Rightarrow F), \mathcal{C})$ sending each object $c\in \mathcal{C}$ to the constant function $(x, y) \mapsto c$. There is also a projection functor $P: (* \Rightarrow F)^{op} \to \mathcal{C}^{op}$, which we can equally think of as a functor $P: (* \Rightarrow F) \to \mathcal{C}$, which assigns $P(x, y) = x$. Prove the Yoneda lemma: $$\text{Hom}_{\text{Fun}(\mathcal{C},\text{Set})}(F, Y(c)) \cong \text{Hom}_{\text{Fun}((*\Rightarrow F), \mathcal{C})}(P, \Delta(c))$$ naturally in $c\in \mathcal{C}$ where $Y(c)$ is the representable functor.
