§ Intuition for limits in category theory
§ A characterization of limits
The theorem that characterizes limits is this:
A category has Finite Limits iff it has all finite products and equalizers. limits have maps out of them.
§ Ravi Vakil's intuition for limits: Take sub things.
 An element of a limit gives one of each of its ingredients. For example,$K[[X]] = \lim_n \text{degree}~n~\text{polynomials}$, since we can geta degree n polynomial for all n, from any power series by truncation.
 limits have maps out of them.
 RAPL: right adjoints preserve limits.
 Limits commute with limits.
 These are mentioned in "Algebraic geometry in the time of Covid: Pseudolecture 2"
 Also, recalling that kernels are limits allows one to remember that we havemaps that go out of a limit.
§ Limits are things you get maps out of.
 product has projections going out of it.

gcd
is less than the things that build it: gdb(a, b) > a, b
since gcd(a, b) < a
and gcd(a, b) < b
.  single point set can be mapped out of.

type Limit f = forall a. f a
§ Limits in Haskell
data Limit (f :: * > *) where
MkLimit :: (forall a. f a) > Limit f
projectOut :: Limit f > f a
projectOut (MkLimit fa) = fa
data Colimit (f:: * > *) where
MkCoLimit :: f a > Colimit f
projectIn :: f a > Colimit f
projectIn = MkCoLimit