Existence of some categories in the category of categories

Question

I'm studying category of categories. I read that when there are categories $A,B$, it is allowed to define the product $A\times B$. Equalizers and coequalizers also exist. However, there are some categories which I don't know how to construct.

1.When one says "Let $A$ be a full subcategory of $B$ such that...", how do we define this $A$ in the category of categories?

2.How do we define "Functors from $A$ to $B$ which preserve finite limits"?

Added

3.I'm reading Lawvere's Functorial Semantics of Algebraic Theories.

At the page 35, he writes

The existence of the category $S_0$ of finite sets, the category $S_1$ of small sets, and the category $S_2$ of large sets now follows, as does the existence of categories $M_0, M_1, M_2$ of monoids.

I don't understand the existence of $M_i$.

Answer 1

For the subcategory terminology only.

---

Recall that a "subset" of a set is just another set obtained by forgetting some of its elements. Likewise, a "subcategory" of a category is just another category obtained by forgetting some of its objects and arrows.

In particular,

    subcategory of 
≡
    is a category with Obj  ⊆ Obj  and Arr  ⊆ Arr 

and

    is a full subcategory of 
≡
    is a subcategory of  obtained by forgetting only some objects

Dully, a co-full/lluf/wide subcategory is one obtained by forgetting only some arrows.

If you want to get universal, then here:

  is a full subcategory of 
≡
  is a subcategory of 
 and
 for any subcategory  of  we have Obj  ⊆ Obj 

Of course, for many, there's a "more categorical" notion. A "subset" is just an injective function; a "substructure" is just an injective homomorphism; and generally, one considers a "subobject" as a monic arrow. So then "subcategory" is just a subobject in some ambient category of categories? For the former cases of, says, sets, fields, etc, the notion of an embedding is used for sub- but for categories there's debate. Anyhow, just some stuff to think about.