For any initial objectsBy this unique isomorphism,X,Y, ∃f∈Mor(X,Y), ∃g∈Mor(Y,X),g•f∈ Mor(X,X) ∧ 1_{X}∈ Mor(X,X) ∴g•f= 1_{X}.

Similarly,f•g= 1_{Y}. Thusfis an isomorphism, unique because Mor(X,Y) is a singleton.∎

Similarly, an object

Such objects exist in many categories, but are not always interesting. For example, in any category of relational systems containing representatives (copies) of all possible ones with a given language:

- Singletons where relations are constantly true, are final objects in categories of one-type systems or algebras; for multi-type systems, final objects are made of one singleton per type.
- The empty set where Boolean constants are false is the initial object.

- Objects are all (
*X*,φ) where*X*is a set and φ:*X*×*K*→*B*; - Mor((
*X*,φ),(*Y*,φ') = {*f*∈*Y*| ∀^{X}*a*∈*X*,∀*k*∈*K*, φ(*a*,*k*) = φ'(*f*(*a*),*k*)}.

For any subset

(There exists a section with image *A*) ⇔ (for any object *N*, Mor(*A*,*N*) =
{*g*_{|A} | *g*∈Mor(*E*,*N*)})
⇒ (any embedding with image *A* is a section).

Let us generalize the concept of embedding to any concrete category *C*, making
it work the same as in categories of relational systems beyond the case of sections
(but unlike sections, it will require checking the whole category).
For any subset *A* of an object *E* of *C*, let To-*A* be the category where

- Objects are all
*C*-morphisms*f*into*A*, conceived as (*F*,*f*) where*F*is any object of*C*,*f*∈ Mor_{C}(*F*,*E*) and Im*f*⊂*A*; - Mor
_{To-A}((*F*,*f*),(*G*,*g*)) = {*h*∈Mor_{C}(*F*,*G*) |*f*=*g*০*h*}

Now a morphism *f* in *C* is an *embedding* if it is a final object of any
To-*A*. Then it is also a final
object of To-(Im *f*). It is an *embedding onto A* if moreover *A* = Im *f*.

All such embeddings, even non-injective ones, are monic : Section ⇒ Embedding ⇒ Monomorphism.

For any object *F*, Mor(*F*,*P*) ≅
∏_{i∈I} Mor(*F*,*E _{i}*)

Any data of

(∀*i*∈*I*, π_{i}∈Mor_{L}(*P*,*E _{i}*)) ⇔
(∀

In the more general case, assuming (Inc), the reverse inclusion not only defines Mor(

For all object *F*, ∏_{i∈I}
Hom (*F*,φ_{i}) :
Mor(*F*,*P*) ↔ ∏_{i∈I} Mor(*F*,*E _{i}*)

Empty products (*I*=∅) are the final objects. The concept of embedding
is a variant of the unary case (itself trivial).

- Objects are all (
*X*,*x*) where*X*is an object of*C*and*x*∈*X* - Mor((
*X*,*x*),(*Y*,*y*)) = {*f*∈Mor(*X*,*Y*) |*f*(*x*)=*y*}.

- If
*C*is the category of*M*-sets for a monoid (*M*,*e*, •) then, seeing*M*as an*M*-set interpreting • as left action, (*M*,*e*) is an initial object of*C'*; initial objects are the (*X*,*x*) where*x*is a free and generating element of*X*. - Conversely, for any initial object (
*M*,*e*) of*C'*(if that exists), there is a unique monoid structure (*M*,*e*,•) with an action on every other object*X*of*C*(beyond • on*M*itself), such that for all objects*X*,*Y*of*C*we have Mor(*X*,*Y*) ⊂ Mor_{M}(*X*,*Y*) and Mor(*M*,*X*) = Mor_{M}(*M*,*X*).

2. Defining ∀

The composition in

The last axiom of monoid,

∀

This monoid (

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1. Morphisms of relational systems and concrete categories4. Model Theory

3.2. Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6. Invertibility and groups

3.7. Categories

3.8.Initial and final objects

3.9. Algebraic terms

3.10. Term algebras

3.11. Integers and recursion

3.12. Presburger Arithmetic