I want to show that the loop space functor $\Omega$ and the suspension functor $\Sigma$ are adjoints from the infinity category of pointed spaces to itself. I guess this means I have to show $Maps(\Sigma X,Y) \simeq Maps(X, \Omega Y)$ where the isomorphism is natural in X and Y.
Let me start by saying that I know how to prove this statement using exponential law (i.e. Hom-tensor adjunction) and the formulae $\Sigma X = S^1 \wedge X, \Omega X = Maps(S^1,X)$. These formulae work in spaces.
But I want to prove this result using the definition/universal properties of suspension and loop space functor given as certain infinity categorical pushouts and pullbacks. That is $\Sigma X := pt \coprod_{X} pt$ and $\Omega Y:= pt \times_Y pt$.
How do I prove the adjunction formula from universal properties?
In my attempt, I computed the two sides of the adjunction "equation" and the computation seems to reduce down to proving $Y \times_{Y^X} Y \simeq \Sigma Y^X.$ I am stuck here.