How is an equalizer a (limit) cone?

Question

I am reading "Category Theory for Computing Science" by Barr & Well. In Ch9 about limits, the authors stated that

Equalizers are also limits

and in particular, they stated (without explanation) that an equalizer is part of a cone (cone as defined in 9.2.1, i.e. a family of arrows from the vertex object indexed by the shape graph G)

But I have a hard time understanding how is the equalizer E above part of a cone? I can understand that a product is a cone where the diagram has no arrows (is discrete).

But how is the above equalizer is a cone? Which is the vertex and which is the diagram?

A further question is then, how is the cone in question a limit cone?

Answer 1

Let $\mathcal{C}$ denote a category (it will become the category in which your equalizer lives).

Consider the categories:

1) $\mathbb{C}$ that has only of two objects $A$ and $B$ (and no morphisms between them) and

2) $\mathbb{D}$ that has only two objects $A$ and $B$ and only two morphisms $f,g:A\to B$ (and of course the necessary identities in both cases).

If you believe that a product is a cone over a functor $F:\mathbb{C}\to\mathcal{C}$, you shouldn't have trouble seeing that an equalizer is a cone over a functor $G:\mathbb{D}\to\mathcal{C}$.