Simplicial Structures in Topology

I.2 Categories 33
Conversely, given ˆf ∈ M∗(X,ΩY ), its adjoint f : I × X → Y is continuous because
I is compact Hausdorff (see Lemma (I.1.39))and f ∈ M∗(ΣX,Y); this allows us to
construct a function
Φ ′ : M∗(X,ΩY) → M∗(ΣX,Y)
such that ΦΦ ′ and Φ ′ Φ be equal to the respective identity functions. In other words,
Φ : M∗(ΣX,Y ) → M∗(X,ΩY )
is a bijection (injective and surjective). For this reason, we say that Σ is left adjoint
to Ω.
Given two functors F,G: C → C ′ ,a natural transformation
η : F → G
is a correspondence that takes each object A ∈ C to a morphism
and such that, for every f ∈ C(A,B),
η(A): FA → GA
G( f )η(A)=η(B)F( f ),
in other words, such that the following diagram is commutative:
F( f )
FA η(A) ��
��
FB
η(B)
GA
��
��
GB
G( f )
Two functors F,G: C → C ′ are equivalent (and we write F . = G)iftherearetwo
natural transformations η : F → G and τ : G → F such that
τη = 1F and ητ = 1G
where 1F and 1G equal the natural transformations given by the identity.
I.2.2 Pushouts
Given two morphisms f : A → B and g: A → C of a category C,apushout of ( f ,g)
is a pair of morphisms f ∈ C(C,D) and g ∈ C(B,D) such that gf = g f and satisfying
the following