I have questions about the proof of Theorem 3.4.12 in Emily Riehl's Category Theory in Context. The theorem states that the colimit of a small diagram $F\colon \mathsf J \to\mathsf C$ can be expressed as a coequalizer between two arrows $c,d\colon\coprod_{f\in\operatorname{mor} J}F(\operatorname{dom} f) \rightarrow \coprod_{j\in\operatorname{ob} J} Fj$:
The theorem: 
We assume that $\mathsf C$ is locally small.
In the first step of the proof, Riehl constructs this diagram: 
The aim of the proof is to show that $C$ defines a colimit of $F.$
By using the fact that the contravariant Yoneda embedding $\mathsf C^{\operatorname{op}} \to \mathsf{Set}^{\mathsf C} $ preserves limits in $\mathsf C^{\operatorname{op}}$, we get the following equalizer diagram in $\mathsf{Set}^{\mathsf{C}}$: 
My questions arise in the next steps. Since contravariant representable functors $\mathsf C(-,X)$ carry colimits in $\mathsf C$ to limits in $\mathsf{Set}$, Riehl claims that we get: 
The first thing I don't understand is how $\mathsf C(d,X)$ becomes $c$ and $\mathsf C(c,X)$ becomes $d$.
The next step of the proof would be to fix $X$. Since the evaluation functor $\operatorname{ev}_X\colon \mathsf{Set}^{\mathsf C} \to \mathsf{Set}$ preserves limits, we have an equalizer diagram in $\mathsf{Set}$.
We use a special case ($\mathsf C= \mathsf{Set}$) of the theorem for limits on the diagram $\mathsf{C}(F-,X)\colon \mathsf J^{\operatorname{op}} \overset{F}{\rightarrow} \mathsf C^{\operatorname{op}} \overset{\mathsf C(-,X)}{\to} \mathsf{Set}$ to conclude that $\lim_{\mathsf{J}^{\operatorname{op}}} \mathsf{C}(F-,X)\cong\mathsf{C}(C,X)$ for each $X \in \mathsf C$. The theorem: 
But how do we know that $c,d$ in (3.4.14) are those arrows whose equalizer is $\lim_{\mathsf{J}^{\operatorname{op}}} \mathsf{C}(F-,X)$?
The end of the proof goes as follows: Let $\operatorname{ob} \mathsf C$ be the maximal discrete subcategory of $\mathsf C$. Since $\operatorname{ob} \mathsf C$ is discrete, the collection of the isomorphisms is an isomorphism in $\mathsf{Set}^{\operatorname{ob} \mathsf C}$. Then, I think, Riehl uses that the forgetful functor $\mathsf{Set}^{\mathsf C} \to \mathsf{Set}^{\operatorname{ob}\mathsf C}$ strictly creates all limits in $\mathsf{Set}$. In fact, she cites the proposition, but it actually needs $\mathsf C$ to be small:
Proposition 3.3.9. If $\mathsf A$ is small, then the forgetful functor $\mathsf C^{\mathsf A} \to \mathsf C^{\operatorname{ob}\mathsf A}$ strictly creates all limits and colimits that exist in $\mathsf C$. These limits are defined objectwise, meaning that for each $a\in \mathsf A$, the evaluation functor $\operatorname{ev}_a\colon \mathsf C^{\mathsf A}\to\mathsf C$ preserves all limits and colimits existing in $\mathsf C$.
Why can we use this proposition?
The proposition gives us that $\mathsf C(C,-)$ is the limit of the $\mathsf J^{\operatorname{op}}$-indexed diagram of covariant functors $\mathsf C(Fj,-)$. We conclude that $C$ is the colimit of $F$ since
Theorem 3.4.7. For any diagram $F\colon \mathsf J \to\mathsf C$ whose colimit exists, there is a natural isomorphism $$ \mathsf{C}(\operatorname{colim}\nolimits_{\mathsf{J}} F,X)\cong \lim\nolimits_{\mathsf{J}^{\operatorname{op}}} \mathsf{C}(F-,X). $$
I hope someone can help me by answering my questions in bold.