===== On category theory basics =====
|[[Foundational temp1b]] $\blacktriangleright$ On category theory basics $\blacktriangleright$ [[On universal morphisms]]|
==== Guide ====
|[[Opposite category]]|

From every category, we can obtain another one by simply flipping the arguments of concatenation.

|[[Isomorphism]]|
|[[Groupoid]]|

|[[Automorphism]]|

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|[[Functor]]|

As far as objects $A,B\in{\bf C}$ are concerned, functors $F$ in ${\bf C}\longrightarrow{\bf D}$ are plain functions, acting via $A\mapsto FA$. For this reason, people sometimes only think of object map when thinking about certain functors, but it's generally very important to remember the arrow map, which in a sense is much finer. Note that the functors object map implicitly also maps the hom-classes "${\bf C}[A,B]\mapsto{\bf C}[FA,FB]$". Now while $F$ doesn't care for any "internal" details of the objects (e.g. the elements, in case the objects are sets), it does indeed care for the terms of the hom-classes!

{{ vorsicht.jpeg?X300}}

|[[Identity functor]]|
|[[Constant functor]]|

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|[[Natural transformation]]|
As opposed to functors defined above, natural transformations indeed look at the "internal" details objects. 

|[[Natural isomorphism]]|
|[[Equivalence of categories]]|
|[[Discrete category]]|
|[[Functor category]]|
|[[Diagonal functor]]|

==== Parents ====
=== Sequel of ===
[[Foundational temp1b]]
=== Related ===
[[Foundational temp1b]]