Natural isomorphism of Left adjoints

dmuthuk
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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).
 
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You have (natural) isos

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

is the composition of (natural) isos an iso?
 
n_bourbaki said:
You have (natural) isos

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

is the composition of (natural) isos an iso?

Yes, I believe so.
 
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