# Natural isomorphism of Left adjoints

## Main Question or Discussion Point

Given two left adjoints $$F,H:\mathcal{C}\to\mathcal{D}$$ of a functor $$G:\mathcal{D}\to\mathcal{C}$$, how do we show that $$F$$ and $$H$$ are naturally isomorphic? This is my idea so far (I am working with the Hom-set defenition of adjunction):

We need to construct a natural isomorphism $$\alpha$$. So, for each $$x\in\mathcal{C}$$, I need a morphism $$\alpha_x:F(x)\to H(x)$$. Suppose we are given the natural isomorphisms $$\varphi:\mbox{Hom}(F-,-)\to\mbox{Hom}(-,G-)$$ and $$\psi:\mbox{Hom}(H-,-)\to\mbox{Hom}(-,G-)$$. Then, I can simply let $$\alpha_x := \varphi_{x,Hx}^{-1}\circ\psi_{x,Hx}(1_{Hx})$$. But, I am stuck here. I don't know how to show that for a given morphism $$f:x\to y$$ in $$\mathcal{C}$$, $$H(f)\circ\alpha_x = \alpha_y\circ F(f)$$.

Related Linear and Abstract Algebra News on Phys.org
You have (natural) isos

(F?,?)-->(?,G?)-->(H?,?)

is the composition of (natural) isos an iso?

You have (natural) isos

(F?,?)-->(?,G?)-->(H?,?)

is the composition of (natural) isos an iso?
Yes, I believe so.