I'm currently learning the deformation of Galois representations by reading Gouvêa's note.
Let $\mathsf{CNL}$ be the category of complete noetherian local rings with (fixed) residue field $\mathbb{F}$. The morphisms are local homomorphisms fixing the residue field. Let $\Lambda \in \mathsf{CNL}$ and denote the full subcategory of $\Lambda$-algebras in $\mathsf{CNL}$ by $\mathsf{CNL}_{\Lambda}$. The full subcategory of artinian objects are denoted by $\mathsf{Ar}_{\Lambda}$.
Let $\mathbf{F}: \mathsf{Ar}_{\Lambda} \rightarrow \mathsf{Set}$ be a functor and $\mathbf{F}_1$ a subfunctor of $\mathbf{F}$. By a "subfunctor", we mean for all $A \in \mathsf{Ar}_{\Lambda}$, $\mathbf{F}_1(A) \subseteq \mathbf{F}(A)$. (More precise definitions here: Subfunctor in nLab)
Now suppose $\mathbf{F}$ and $\mathbf{F}_1$ are pro-representable, i.e. they are represented by object in $\mathsf{CNL}_{\Lambda}$. Let $R, R_1 \in \mathsf{CNL}_{\Lambda}$ represent $\mathbf{F}$ and $\mathbf{F}_1$ accordingly, then one can show that $R_1$ is a quotient of $R$. Say $R_1 = R/I$.
Let $A \in \mathsf{Ar}_{\Lambda}$ and $a \in \mathbf{F}_1(A) \subseteq \mathbf{F}(A)$ fixed. Then for $a$,
- By representability of $\mathbf{F}_1$, there exists a corresponding homomorphism $\phi_{1,a}: R_1 \rightarrow A$. (Via $\mathbf{F}_1(A) \cong \mathrm{Hom}(R_1, A)$.)
- By representability of $\mathbf{F}$, there exists a corresponding homomorphism $\phi_{a}: R \rightarrow A$. (Via $\mathbf{F}(A) \cong \mathrm{Hom}(R, A)$.)
My main question is: Are the two homomorphisms $\phi_{1,a}$ and $\phi_{a}$ "coherent"?
To be more specific, let $\pi: R \rightarrow R_1=R/I$ be the natural homomorphism. Is the composition $$ R \stackrel{\pi}{\longrightarrow} R/I \stackrel{\phi_{1,a}}{\longrightarrow} A $$ EXACTLY the homomorphism $\phi_{a}$?
I feel like this is correct but I have spent quite a long time struggling for a neat proof but failed. So I'm sitll doubting whether this is really true or not?
Thank you all for commenting and answering!
The Yoneda embedding associates to any natural transformation $\text{Hom}(R_1,-)\to \text{Hom}(R,-)$ a ring morphism $\pi:R \to R_1$ such that for any $T$ $$\text{Hom}(R_1,T)\to \text{Hom}(R,T),\quad\phi\mapsto \phi\circ \pi$$ is the morphism induced by the natural transformation. Choose an arbitrary $a\in F_1(A)$ given by $\phi_{1,a}\in \text{Hom}(R_1,A).$ Then its image under the inclusion of the subfunctor, by definition of the induced ring morphism $R\to R_1$, is given by the image of $\phi_{1,a}$ under $\text{Hom}(R_1,T)\to \text{Hom}(R,T)$, i.e its $\phi_{1,a}\circ \pi$. Hence $\phi_{1,a}\circ \pi=\phi_a$ to use your notation.